Patent Publication Number: US-2022232512-A1

Title: Positioning method and apparatus based on binary periodic sequence

Description:
TECHNICAL FIELD 
     The present application relates to the field of communications, and in particular to a positioning method and apparatus based on a binary periodic sequence. 
     BACKGROUND 
     The binary sequence has been widely used in the engineering field, especially in wireless communications and navigation positioning. 
     Let F q  be a finite field of q elements, and let q be a prime power (power of a prime number). A set of all nonzero elements in F q  is denoted as F q *. Let F be a set of binary sequences, each binary sequence having a length of n. Then, for each sequence s=(s 0 , s 1 , . . . , s n-1 )∈F, where s i  ∈{1,−1}, it is defined that an auto-correlation coefficient of s at delay t(1≤t≤n−1) is: 
     
       
         
           
             
               
                 A 
                 t 
               
               ⁡ 
               
                 ( 
                 s 
                 ) 
               
             
             := 
             
               
                 ∑ 
                 
                   i 
                   = 
                   0 
                 
                 
                   n 
                   - 
                   1 
                 
               
               ⁢ 
               
                 
                   s 
                   i 
                 
                 ⁢ 
                 
                   s 
                   
                     i 
                     + 
                     t 
                   
                 
               
             
           
         
       
     
     where i+t=(i+t) mod n. Then, for two different sequences u=(u 0 , u q , . . . , u n-1 ) and v=(v 0 , v 1 , . . . , v n-1 )∈F, u i , v j ∈{1, −1}, define a cross-correlation coefficient of u and v at delay t(1≤t≤n−1) as: 
     
       
         
           
             
               
                 C 
                 t 
               
               ⁡ 
               
                 ( 
                 
                   u 
                   , 
                   v 
                 
                 ) 
               
             
             := 
             
               
                 ∑ 
                 
                   i 
                   = 
                   0 
                 
                 
                   n 
                   - 
                   1 
                 
               
               ⁢ 
               
                 
                   u 
                   i 
                 
                 ⁢ 
                 
                   v 
                   
                     i 
                     + 
                     t 
                   
                 
               
             
           
         
       
     
     The correlation of a sequence cluster   is defined as: 
     
       
         
           
             
               Cor 
               ⁡ 
               
                 ( 
                 ℱ 
                 ) 
               
             
             := 
             
               max 
               ⁢ 
               
                 { 
                 
                   
                     
                       max 
                       
                         
                           s 
                           ∈ 
                           ℱ 
                         
                         , 
                         
                           1 
                           ≤ 
                           t 
                           ≤ 
                           
                             n 
                             - 
                             1 
                           
                         
                       
                     
                     ⁢ 
                     
                       { 
                       
                         
                           A 
                           t 
                         
                         ⁡ 
                         
                           ( 
                           s 
                           ) 
                         
                       
                       } 
                     
                   
                   , 
                   
                     
                       max 
                       
                         
                           
                             u 
                             ≠ 
                             v 
                           
                           ∈ 
                           ℱ 
                         
                         , 
                         
                           0 
                           ≤ 
                           t 
                           ≤ 
                           
                             n 
                             - 
                             1 
                           
                         
                       
                     
                     ⁢ 
                     
                       { 
                       
                         
                           C 
                           t 
                         
                         ⁡ 
                         
                           ( 
                           
                             u 
                             , 
                             v 
                           
                           ) 
                         
                       
                       } 
                     
                   
                 
                 } 
               
             
           
         
       
     
     Among the known sequences, the Gold sequence is widely applied due to better correlation. The Gold sequence is a pseudo-random sequence with good characteristics. There are various methods for constructing the Gold sequence. One of the typical construction methods is to make use of a trace function. Let q=2 n  and let γ be a generator of F q *. Then the Gold sequence may be represented by: 
     
       
         
           
             
               { 
               
                 
                   
                     ( 
                     
                       - 
                       1 
                     
                     ) 
                   
                   
                     Tr 
                     ⁡ 
                     
                       ( 
                       
                         a 
                         + 
                         b 
                       
                       ) 
                     
                   
                 
                 , 
                 
                   
                     ( 
                     
                       - 
                       1 
                     
                     ) 
                   
                   
                     Tr 
                     ⁡ 
                     
                       ( 
                       
                         
                           a 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           γ 
                         
                         + 
                         
                           b 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             γ 
                             3 
                           
                         
                       
                       ) 
                     
                   
                 
                 , 
                 … 
                 ⁢ 
                 
                     
                 
                 , 
                 
                   
                     
                       ( 
                       
                         - 
                         1 
                       
                       ) 
                     
                     
                       Tr 
                       ( 
                       
                         
                           a 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             γ 
                             
                               q 
                               - 
                               2 
                             
                           
                         
                         + 
                         
                           b 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             γ 
                             
                               
                                 ( 
                                 
                                   q 
                                   - 
                                   2 
                                 
                                 ) 
                               
                               3 
                             
                           
                         
                       
                     
                   
                   | 
                   a 
                 
                 , 
                 b 
                 , 
                 
                   ∈ 
                   
                     F 
                     q 
                   
                 
               
               } 
             
             ; 
           
         
       
     
     where Tr is the trace function from F q  to F 2 . Since the correlation of the Gold sequence depends on rational points on a curve y 2 +y=c 3 +dx, the correlation of the Gold sequence may be estimated with a Hasse-Weil bound. 
     However, no solution has been proposed yet for the following problem in the related art: although the constructed binary sequence has a good correlation mean value, the peak of correlation coefficient of the positioning signal generated in the positioning application is large. 
     SUMMARY 
     Embodiments of the present disclosure provide a positioning method and apparatus based on a binary periodic sequence to solve at least the following problem in the related art: although the constructed binary sequence has a good correlation mean value, the peak of correlation coefficient of the positioning signal generated in the positioning application is large. 
     According to an embodiment of the disclosure, there is provided a positioning method based on a binary periodic sequence, including: selecting one polynomial from each equivalence class of a quadratic polynomial set (set of quadratic polynomials) S to determine a set T; constructing a binary periodic sequence cluster according to the set T; generating a positioning signal according to the binary periodic sequence cluster; and performing positioning processing according to the positioning signal. 
     According to another aspect of the embodiment of the present disclosure, there is further provided a positioning apparatus based on a binary periodic sequence, including: a first selection module configured to select one polynomial from each equivalence class of a quadratic polynomial set S to determine a set T; a construction module configured to construct a binary periodic sequence cluster according to the set T; a generating module configured to generate a positioning signal according to the binary periodic sequence cluster; and a positioning processing module configured to perform positioning processing according to the positioning signal. 
     According to still another embodiment of the disclosure, there is further provided a storage medium having a computer program stored thereon, wherein the computer program is configured to be executed to cause steps of any one of the above method embodiments to be implemented. 
     According to still another embodiment of the disclosure, there is further provided an electronic apparatus, including a memory and a processor, wherein the memory has a computer program stored thereon, and the processor is configured to execute the computer program to implement steps of any of the method embodiments as described above. 
     According to the present disclosure, a polynomial determination set T is selected from each equivalence class of a quadratic polynomial set S; a binary periodic sequence cluster is constructed according to the set T; a positioning signal is generated according to the binary periodic sequence cluster; and positioning processing is performed according to the positioning signal, thereby solving the following problem in the related art: although the constructed binary sequence has a good correlation mean value, the peak of correlation coefficient of the positioning signal generated in the positioning application is large. Since the constructed binary sequence with a new length has good correlation coefficient, the peak of positioning correlation coefficient generated in the positioning application is relatively small. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
       The drawings described herein are intended to provide a further understanding of the present disclosure, and are intended to be a part of the present disclosure. The exemplary embodiments of the present disclosure and the description thereof are for explaining the present disclosure and do not constitute an undue limitation of the present disclosure. In the drawings: 
         FIG. 1  is a block diagram showing a hardware structure of a mobile terminal used in a positioning method based on a binary periodic sequence according to an embodiment of the present disclosure; 
         FIG. 2  is a flowchart of a positioning method based on a binary periodic sequence according to an embodiment of the present disclosure; 
         FIG. 3  is a schematic diagram showing auto-correlation values of a multiplicative group sequence  1  according to an embodiment of the present disclosure; 
         FIG. 4  is a schematic diagram showing auto-correlation values of a multiplicative group sequence  2  according to an embodiment of the present disclosure; 
         FIG. 5  is a schematic diagram showing cross-correlation values of a multiplicative group sequence according to an embodiment of the present disclosure; 
         FIG. 6  is a schematic diagram showing auto-correlation values of a multiplicative group irreducible polynomial sequence  1  according to an embodiment of the disclosure; 
         FIG. 7  is a schematic diagram showing auto-correlation values of a multiplicative group irreducible polynomial sequence  2  according to an embodiment of the disclosure; 
         FIG. 8  is a schematic diagram showing cross-correlation values of a multiplicative group irreducible polynomial sequence according to an embodiment of the disclosure; 
         FIG. 9  is a schematic diagram showing auto-correlation values of an additive group irreducible polynomial sequence  1  according to an embodiment of the disclosure; 
         FIG. 10  is a schematic diagram showing auto-correlation values of an additive group irreducible polynomial sequence  2  according to an embodiment of the disclosure; 
         FIG. 11  is a schematic diagram showing cross-correlation values of an additive group irreducible polynomial sequence according to an embodiment of the disclosure; 
         FIG. 12  is a schematic diagram showing auto-correlation values of a positioning reference signal generated according to an embodiment of the present disclosure; 
         FIG. 13  is a schematic diagram showing cross-correlation values of two positioning reference signals generated according to an embodiment strip the present disclosure; and 
         FIG. 14  is a block diagram of a positioning apparatus based on a binary periodic sequence according to an embodiment of the present disclosure. 
     
    
    
     DETAIL DESCRIPTION OF EMBODIMENTS 
     The disclosure will be described in detail below with reference to the drawings in conjunction with the embodiments. It should be noted that embodiments of the disclosure and features therein may be combined with each other in any manner as long as they are not contradictory. 
     It should be also noted that terms “first”, “second”, and the like in the description, claims and drawings of the disclosure are used for the purpose of distinguishing similar objects instead of indicating a specific order or sequence. In addition, sets T 1 , T 1 ′, T 2 , T 2 ′ and the like are examples of the above set T, sets S 1 , S 1 ′, S 2 ′ and the like are examples of the above quadratic polynomial set S, and    1 ,   1 ′,   2 ,   2 ″ and the like are examples of the above binary periodic sequence cluster. 
     Embodiment 1 
     The method embodiment provided in Embodiment 1 of the present disclosure may be implemented in a mobile terminal, a computer terminal or the like. Taking running on a mobile terminal as an example,  FIG. 1  is a block diagram showing a hardware structure of a mobile terminal used in a positioning method based on a binary periodic sequence according to an embodiment of the present disclosure. As shown in  FIG. 1 , a mobile terminal  10  may include one or more (only one is shown in  FIG. 1 ) processors  102  (which may include, but are not limited to, microprocessor units (MCUs), programmable logic devices such as FPGAs or other processing devices), and a memory  104  configured to store data. Optionally, the mobile terminal may further include a transmission device  106  for communication functions and an input/output device  108 . It will be understood by those ordinary skilled in the art that the structure shown in  FIG. 1  is merely illustrative, and does not form any limitation to the structure of the above mobile terminal. For example, the mobile terminal  10  may include more or fewer components than those shown in  FIG. 1 , or have a different configuration than that shown in  FIG. 1 . 
     The memory  104  may be configured to store a computer program, for example, a software program and a module of application software, such as a computer program corresponding to the positioning method based on a binary periodic sequence in the embodiments of the present disclosure, and the processor  102  executes the computer program stored in the memory  104  to perform various functional applications and data processing, that is, implement the above method. The memory  104  may include a high speed random access memory and may also include a non-volatile memory such as one or more magnetic storage device, flash memory, or other non-volatile solid state memory. In some examples, the memory  104  may further include a memory remotely located relative to the processor  102 , which may be connected to the mobile terminal  10  via a network. Examples of such networks include, but are not limited to, the Internet, intranets, local area networks, mobile communication networks, and combinations thereof. 
     The transmission device  106  is configured to receive or transmit data via a network. Specific examples of such networks may include a wireless network provided by a communication provider of the mobile terminal  10 . In an example, the transmission device  106  includes a Network Interface Controller (NIC) that may be connected to another network device through a base station to communicate with the Internet. In an example, the transmission device  106  may be a Radio Frequency (RF) module configured to communicate with the Internet wirelessly. 
     Based on the above mobile terminal, this embodiment provides a positioning method based on a binary periodic sequence.  FIG. 2  is a flowchart of a positioning method based on a binary periodic sequence according to an embodiment of the present disclosure. As shown in  FIG. 2 , the flow includes steps S 202  to S 208 . 
     At step S 202 , select one polynomial from each equivalence class of a quadratic polynomial set S to determine a set T. 
     In the above step S 202 , specifically, polynomials are selected from the respective equivalence classes of the quadratic polynomial set S, and the set T is determined according to the respective selected polynomials. 
     At step S 204 , construct a binary periodic sequence cluster according to the set T. 
     At step S 206 , generate a positioning signal according to the binary periodic sequence cluster. 
     At step S 208 , perform positioning processing according to the positioning signal. 
     Through the above steps S 202  to S 208 , the following problem in the related art is solved: although the constructed binary sequence has a good correlation mean value, the peak of correlation coefficient of the positioning signal generated in the positioning application is large. Since the constructed binary sequence with a new length has good correlation coefficient, the peak of positioning correlation coefficient generated in the positioning application is relatively small. 
     In an embodiment of the present disclosure, the above step S 206  may specifically include:
         selecting a positioning sequence from the binary periodic sequence cluster according to a preset parameter;   taking a truncated sequence with a preset length from the positioning sequence and determining the truncated sequence as a positioning reference sequence; and   generating the positioning signal by modulating the positioning reference sequence.       

     In an embodiment of the present disclosure, the above step S 208  may specifically include: 
     mapping the positioning signal to an antenna port at a transmitting end for transmission. The positioning signal is configured to instruct a receiving end, after receiving the positioning signal, to: perform cross-correlation to obtain an arrival time of the positioning signal, determine a distance from the transmitting end according to the arrival time, and perform positioning (on the transmitting end) according to the distance. 
     In an embodiment of the present disclosure, before selecting one polynomial from each equivalence class of the quadratic polynomial set S to determine the set T, select one generator γ from F q *, where F q  is a finite field containing q elements, q is an odd prime or an odd prime power (i.e., a power of an odd prime number), and F q * is a set of all nonzero elements in F q . 
     In an embodiment, the above step S 202  may specifically include:
         selecting one polynomial f(x) (i.e., [x 2 +ax+b]) from each equivalence class of a quadratic polynomial set S 1 , and combining the polynomial f(x) with x−1 to form a set T 1 :       

     
       
         
           
             
               T 
               1 
             
             = 
             
               
                 { 
                 
                   x 
                   - 
                   1 
                 
                 } 
               
               ⋃ 
               
                 { 
                 
                   
                     
                       x 
                       2 
                     
                     + 
                     
                       a 
                       ⁢ 
                       x 
                     
                     + 
                     b 
                   
                   ∈ 
                   
                     
                       S 
                       1 
                     
                     ⁢ 
                     
                       ⁢ 
                       
                         
                           [ 
                           
                             
                               x 
                               2 
                             
                             + 
                             
                               a 
                               ⁢ 
                               x 
                             
                             + 
                             b 
                           
                           ] 
                         
                         ⁢ 
                         
                           
 
                         
                         ⁢ 
                         are 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         equivalence 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         classes 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         distinct 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         to 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         each 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         other 
                       
                     
                   
                 
                 } 
               
             
           
         
       
       
         
           
             
               where 
               ⁢ 
               
                   
               
               ⁢ 
               
                 S 
                 1 
               
             
             = 
             
               
                 { 
                 
                   
                     
                       x 
                       2 
                     
                     + 
                     ax 
                     + 
                     b 
                   
                   , 
                   
                     a 
                     ∈ 
                     
                       
                         F 
                         q 
                         * 
                       
                       ⁢ 
                     
                   
                   , 
                   
                     b 
                     ∈ 
                     
                       F 
                       q 
                     
                   
                 
                 } 
               
               ⁢ 
               \ 
               ⁢ 
               
                 
                   { 
                   
                     
                       
                         ( 
                         
                           x 
                           - 
                           a 
                         
                         ) 
                       
                       2 
                     
                     , 
                     
                       a 
                       ∈ 
                       
                         
                           F 
                           q 
                           * 
                         
                         ⁢ 
                       
                     
                   
                   } 
                 
                 . 
               
             
           
         
       
     
     In an embodiment of the present disclosure, the above step S 204  may specifically include: 
     constructing a binary periodic sequence cluster    1  from the generator γ and the set T 1 : 
     
       
         
           
             
               
                 ℱ 
                 1 
               
               = 
               
                 { 
                 
                   
                     s 
                     f 
                   
                   , 
                   
                     
                       f 
                       ⁡ 
                       
                         ( 
                         x 
                         ) 
                       
                     
                     ∈ 
                     
                       T 
                       1 
                     
                   
                 
                 } 
               
             
             ; 
           
         
       
       
         
           
             
               
                 s 
                 f 
               
               = 
               
                 { 
                 
                   
                     η 
                     ⁡ 
                     
                       ( 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           1 
                           ) 
                         
                       
                       ) 
                     
                   
                   , 
                   
                     η 
                     ⁡ 
                     
                       ( 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           γ 
                           ) 
                         
                       
                       ) 
                     
                   
                   , 
                   … 
                   ⁢ 
                   
                       
                   
                   , 
                   
                     η 
                     ⁡ 
                     
                       ( 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           
                             γ 
                             
                               q 
                               - 
                               2 
                             
                           
                           ) 
                         
                       
                       ) 
                     
                   
                 
                 } 
               
             
             ; 
           
         
       
       
         
           
             
               η 
               ⁡ 
               
                 ( 
                 α 
                 ) 
               
             
             = 
             
               { 
               
                 
                   
                     
                       1 
                       , 
                     
                   
                   
                     
                       if 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       α 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       is 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       a 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       nonzero 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       square 
                     
                   
                 
                 
                   
                     
                       - 
                       1. 
                     
                   
                   
                     
                       if 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       α 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       is 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       a 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       non 
                       ⁢ 
                       
                         - 
                       
                       ⁢ 
                       square 
                     
                   
                 
                 
                   
                     
                       0 
                       , 
                     
                   
                   
                     
                       
                         if 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         α 
                       
                       = 
                       0 
                     
                   
                 
               
             
           
         
       
     
     where η is a quadratic multiplicative character from F q  to C*, C is a complex number set excluding 0, and α is any element in F q *. 
     Optionally, each sequence in the binary periodic sequence cluster F 1  has a length of q−1; 
     the binary periodic sequence cluster    1  has a size of q; and 
     if q≥17 and is an odd prime power, a correlation upper limit of the binary periodic sequence cluster    1  is Cor(   1 )≤6+└2√{square root over (q)}┘. 
     Furthermore, the above step S 202  may further specifically include: 
     in the case where x 2 +ax+b in S 1  is an irreducible polynomial, 
     
       
         
           
             
               
                 S 
                 1 
                 ′ 
               
               = 
               
                 { 
                 
                   
                     
                       x 
                       2 
                     
                     + 
                     ax 
                     + 
                     b 
                   
                   , 
                   
                     a 
                     ∈ 
                     
                       
                         F 
                         q 
                         * 
                       
                       ⁢ 
                     
                   
                   , 
                   
                     b 
                     ∈ 
                     
                       F 
                       q 
                     
                   
                   , 
                   
                     
                       x 
                       2 
                     
                     + 
                     ax 
                     + 
                     
                       b 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       is 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       an 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       irreducible 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       polynomial 
                     
                   
                 
                 ⁢ 
                 
                     
                 
                 } 
               
             
             ; 
           
         
       
         
         
           
             selecting one polynomial f(x) from each equivalence class of a quadratic polynomial set S 1 ′, and combining the polynomial f(x) with x−1 to form a set T 1 ′: 
           
         
       
    
     
       
         
           
             
               T 
               1 
               ′ 
             
             = 
             
               
                 { 
                 
                   x 
                   - 
                   1 
                 
                 } 
               
               ⋃ 
               
                 { 
                 
                   
                     
                       x 
                       2 
                     
                     + 
                     
                       a 
                       ⁢ 
                       x 
                     
                     + 
                     b 
                   
                   ∈ 
                   
                     
                       S 
                       1 
                       ′ 
                     
                     ⁢ 
                     
                       ⁢ 
                       
                         
                           [ 
                           
                             
                               x 
                               2 
                             
                             + 
                             
                               a 
                               ⁢ 
                               x 
                             
                             + 
                             b 
                           
                           ] 
                         
                         ⁢ 
                         
                           
 
                         
                         ⁢ 
                         are 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         equivalence 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         classes 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         distinct 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         to 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         each 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         other 
                       
                     
                   
                 
                 } 
               
             
           
         
       
     
     are 
     Correspondingly, the step S 204  may specifically further include: 
     constructing a binary periodic sequence cluster    1 ′ from the generator γ and the set T 1 ′: 
     
       
         
           
             
               
                 ℱ 
                 1 
                 ′ 
               
               = 
               
                 { 
                 
                   
                     s 
                     f 
                   
                   , 
                   
                     
                       f 
                       ⁡ 
                       
                         ( 
                         x 
                         ) 
                       
                     
                     ∈ 
                     
                       T 
                       1 
                       ′ 
                     
                   
                 
                 } 
               
             
             ; 
           
         
       
       
         
           
             
               
                 s 
                 f 
               
               = 
               
                 { 
                 
                   
                     η 
                     ⁡ 
                     
                       ( 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           1 
                           ) 
                         
                       
                       ) 
                     
                   
                   , 
                   
                     η 
                     ⁡ 
                     
                       ( 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           γ 
                           ) 
                         
                       
                       ) 
                     
                   
                   , 
                   … 
                   ⁢ 
                   
                       
                   
                   , 
                   
                     η 
                     ⁡ 
                     
                       ( 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           
                             γ 
                             
                               q 
                               - 
                               2 
                             
                           
                           ) 
                         
                       
                       ) 
                     
                   
                 
                 } 
               
             
             ; 
           
         
       
       
         
           
             
               η 
               ⁡ 
               
                 ( 
                 α 
                 ) 
               
             
             = 
             
               { 
               
                 
                   
                     
                       1 
                       , 
                     
                   
                   
                     
                       if 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       α 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       is 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       a 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       nonzero 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       square 
                     
                   
                 
                 
                   
                     
                       - 
                       1. 
                     
                   
                   
                     
                       if 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       α 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       is 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       a 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       non 
                       ⁢ 
                       
                         - 
                       
                       ⁢ 
                       square 
                     
                   
                 
                 
                   
                     
                       0 
                       , 
                     
                   
                   
                     
                       
                         if 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         α 
                       
                       = 
                       0 
                     
                   
                 
               
             
           
         
       
         
         
           
             where η is a quadratic multiplicative character from F q  to C*, C is a complex number set excluding 0, and α is any element in F q *. 
           
         
       
    
     Optionally, each sequence in the binary periodic sequence cluster    1 ′ has a length of q−1;
         the binary periodic sequence cluster    1 ′ has a size of ½(q−1); and   if q≥11 and is an odd prime power, a correlation coefficient upper limit of the binary periodic sequence cluster    1 ′ is Cor(   1 ′)≤└2√{square root over (q)}┘.       

     In an embodiment of the present disclosure, before selecting one polynomial from each equivalence class of the quadratic polynomial set S to determine the set T, equivalence class division is performed on the polynomials in the quadratic polynomial set S, which includes:
         determining, if and only if there exists β∈F q * such that a 1 =βa 2 , b1=β 2 b 2 , an equivalence relation between x 2 +a 1 x+b 1  and x 2 +a 2 x+b 2 , and determining an equivalence class of the quadratic polynomial set S to be [x 2 +a 1 x+b 1 ].       

     In another embodiment, the above step S 202  may specifically include:
         selecting one polynomial f(x) from each equivalence class of a quadratic polynomial set S 2  and combining the polynomial f(x) with x to form a set T 2 :       

     
       
         
           
             
               
                 T 
                 2 
               
               = 
               
                 
                   { 
                   x 
                   } 
                 
                 ⋃ 
                 
                   { 
                   
                     
                       
                         
                           x 
                           2 
                         
                         + 
                         
                           a 
                           ⁢ 
                           x 
                         
                         + 
                         b 
                       
                       ∈ 
                       
                         S 
                         2 
                       
                     
                     , 
                     
                       
                         [ 
                         
                           
                             x 
                             2 
                           
                           + 
                           
                             a 
                             ⁢ 
                             x 
                           
                           + 
                           b 
                         
                         ] 
                       
                       ⁢ 
                       
                         
 
                       
                       ⁢ 
                       are 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       equivalence 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       classes 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       distinct 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       to 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       each 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       other 
                     
                   
                   } 
                 
               
             
             ; 
           
         
       
         
         
           
             S 2 ={x 2  ax+b, a,b∈F q }\{(x−a) 2 , a∈F q }, where F q  is a finite field containing q elements, and q is an odd prime. 
           
         
       
    
     Correspondingly, the step S 204  may specifically include:
         constructing a binary periodic sequence cluster    2  from the set T 2 :       

     
       
         
           
             
               
                 ℱ 
                 2 
               
               = 
               
                 { 
                 
                   
                     u 
                     f 
                   
                   , 
                   
                     
                       f 
                       ⁡ 
                       
                         ( 
                         x 
                         ) 
                       
                     
                     ∈ 
                     
                       T 
                       2 
                     
                   
                 
                 } 
               
             
             ; 
           
         
       
       
         
           
             
               
                 u 
                 f 
               
               = 
               
                 { 
                 
                   
                     η 
                     ⁡ 
                     
                       ( 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           0 
                           ) 
                         
                       
                       ) 
                     
                   
                   , 
                   
                     η 
                     ⁡ 
                     
                       ( 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           1 
                           ) 
                         
                       
                       ) 
                     
                   
                   , 
                   … 
                   ⁢ 
                   
                       
                   
                   , 
                   
                     η 
                     ⁡ 
                     
                       ( 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           
                             q 
                             - 
                             1 
                           
                           ) 
                         
                       
                       ) 
                     
                   
                 
                 } 
               
             
             ; 
           
         
       
       
         
           
             
               η 
               ⁡ 
               
                 ( 
                 α 
                 ) 
               
             
             = 
             
               { 
               
                 
                   
                     
                       1 
                       , 
                     
                   
                   
                     
                       if 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       α 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       is 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       a 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       nonzero 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       square 
                     
                   
                 
                 
                   
                     
                       - 
                       1. 
                     
                   
                   
                     
                       if 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       α 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       is 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       a 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       non 
                       ⁢ 
                       
                         - 
                       
                       ⁢ 
                       square 
                     
                   
                 
                 
                   
                     
                       0 
                       , 
                     
                   
                   
                     
                       
                         if 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         α 
                       
                       = 
                       0 
                     
                   
                 
               
             
           
         
       
         
         
           
              where η is a quadratic multiplicative character from F q  to C*, C is a complex number set excluding 0, and α is any element in F q *. 
           
         
       
    
     Optionally, each sequence in the binary periodic sequence cluster    2  has a length of q;
         the binary periodic sequence cluster    2  has a size of q; and   if q≥17 and is an odd prime, a correlation coefficient upper limit of the binary periodic sequence cluster    2  is Cor(   2 )≤5+└2√{square root over (q)}┘.       

     In an embodiment, the above step S 202  may further specifically include:
         in the case where the x 2 +ax+b in S 2  is an irreducible polynomial,       

     
       
         
           
             
               
                 S 
                 2 
                 ′ 
               
               = 
               
                 { 
                 
                   
                     
                       x 
                       2 
                     
                     + 
                     
                       a 
                       ⁢ 
                       x 
                     
                     + 
                     b 
                   
                   , 
                   a 
                   , 
                   
                     b 
                     ∈ 
                     
                       F 
                       q 
                     
                   
                   , 
                   
                     
                       x 
                       2 
                     
                     + 
                     
                       a 
                       ⁢ 
                       x 
                     
                     + 
                     
                       b 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       is 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       an 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       irreducible 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       polynomial 
                     
                   
                 
                 ⁢ 
                 
                     
                 
                 } 
               
             
             ; 
           
         
       
         
         
           
             selecting one polynomial f(x) from each equivalence class of a quadratic polynomial set S 2 ′ and combining the polynomial f(x) with x to form a set T 2 ′: 
           
         
       
    
     
       
         
           
             
               T 
               2 
               ′ 
             
             = 
             
               
                 { 
                 x 
                 } 
               
               ⋃ 
               
                 
                   { 
                   
                     
                       
                         
                           x 
                           2 
                         
                         + 
                         ax 
                         + 
                         b 
                       
                       ∈ 
                       
                         S 
                         2 
                         ′ 
                       
                     
                     , 
                     
                       
                         [ 
                         
                           
                             x 
                             2 
                           
                           + 
                           ax 
                           + 
                           b 
                         
                         ] 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       are 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       equivalence 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       classes 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       distinct 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       to 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       each 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       other 
                     
                   
                   } 
                 
                 . 
               
             
           
         
       
     
     Correspondingly, the step S 204  may further include:
         constructing a binary periodic sequence cluster    2 ′ from the set T 2 ′       

     
       
         
           
             
               
                 ℱ 
                 2 
                 ′ 
               
               = 
               
                 { 
                 
                   
                     u 
                     f 
                   
                   , 
                   
                     
                       f 
                       ⁡ 
                       
                         ( 
                         x 
                         ) 
                       
                     
                     ∈ 
                     
                       T 
                       2 
                       ′ 
                     
                   
                 
                 } 
               
             
             ; 
           
         
       
       
         
           
             
               
                 u 
                 f 
               
               = 
               
                 { 
                 
                   
                     η 
                     ⁡ 
                     
                       ( 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           0 
                           ) 
                         
                       
                       ) 
                     
                   
                   , 
                   
                     η 
                     ⁡ 
                     
                       ( 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           0 
                           ) 
                         
                       
                       ) 
                     
                   
                   , 
                   … 
                   ⁢ 
                   
                       
                   
                   , 
                   
                     η 
                     ⁡ 
                     
                       ( 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           
                             q 
                             - 
                             1 
                           
                           ) 
                         
                       
                       ) 
                     
                   
                 
                 } 
               
             
             ; 
           
         
       
       
         
           
             
               η 
               ⁡ 
               
                 ( 
                 α 
                 ) 
               
             
             = 
             
               { 
               
                 
                   
                     
                       1 
                       , 
                     
                   
                   
                     
                       if 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       α 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       is 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       a 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       nonzero 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       square 
                     
                   
                 
                 
                   
                     
                       - 
                       1. 
                     
                   
                   
                     
                       if 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       α 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       is 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       a 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       non 
                       ⁢ 
                       
                         - 
                       
                       ⁢ 
                       square 
                     
                   
                 
                 
                   
                     
                       0 
                       , 
                     
                   
                   
                     
                       
                         if 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         α 
                       
                       = 
                       0 
                     
                   
                 
               
             
           
         
       
         
         
           
              where η is a quadratic multiplicative character from F q  to C*, C is a complex number set excluding 0, and α is any element in F q *. 
           
         
       
    
     Optionally, each sequence in the binary periodic sequence cluster    2 ′ has a length of q; 
     the binary periodic sequence cluster    2 ′ has a size of (q−1)/2; and 
     if q≥7 and is an odd prime, a correlation coefficient upper limit of the binary periodic sequence cluster    2 ′ is Cor(   4 )≤1+└2√{square root over (q)}┘. 
     In an embodiment of the present disclosure, before selecting one polynomial from each equivalence class of the quadratic polynomial set S to determine the set T, the step of performing equivalence class division on the polynomials in the quadratic polynomial set S, may include: 
     determining, if and only if there exists β∈F q , a 1 =2β+a 2 ,b 1 =β 2 +βa+b 2 , an equivalence relation between x 2 +a 1 x+b 1  and x 2 +a 2 x+b 2 , and determining an equivalence class of the quadratic polynomial set S to be [x 2 +a 1 x+b 1 ]. 
     An embodiment of the present disclosure provides a method for designing a binary periodic sequence based on prime characteristics, in which a length of the sequence may be an odd prime (or odd prime−1) or a prime power (or prime power −1), and has a better correlation coefficient. The binary periodic sequence is a special sequence, i.e., a periodic sequence that takes only two nonzero values. 
     Example 1 
     Based on a multiplicative group structure of a prime characteristic finite field, q is an odd prime or an odd prime power. Consider a monic quadratic polynomial set: 
     
       
         
           
             
               
                 S 
                 1 
               
               = 
               
                 
                   { 
                   
                     
                       
                         x 
                         2 
                       
                       + 
                       
                         a 
                         ⁢ 
                         x 
                       
                       + 
                       b 
                     
                     , 
                     
                       a 
                       ∈ 
                       
                         F 
                         q 
                         * 
                       
                     
                     , 
                     
                       b 
                       ∈ 
                       
                         F 
                         q 
                       
                     
                   
                   } 
                 
                 ∖ 
                 
                   { 
                   
                     
                       
                         ( 
                         
                           x 
                           - 
                           a 
                         
                         ) 
                       
                       2 
                     
                     , 
                     
                       a 
                       ∈ 
                       
                         F 
                         q 
                         * 
                       
                     
                   
                   } 
                 
               
             
             ; 
           
         
       
         
         
           
             the number of the sets is q(q−1)−(q−1)=(q−1) 2 . 
           
         
       
    
     Define an equivalence relation in S 1 : x 2 +a 1 x+b 1 ˜x 2 +a 2 x+b 2 , if and only if there exists β∈F q * such that a 1 =βa 2 , b1=β 2 b 2 , there are exactly q−1 elements in each equivalence class. Thus, there is exactly q−1 equivalence classes, denoted as [x 2 +a 1 x+b 1 ]. 
     Select one polynomial in each equivalence class, and combine the selected polynomials with x−1 to form a set a set T 1 : 
     
       
         
           
             
               T 
               1 
             
             = 
             
               
                 { 
                 
                   x 
                   - 
                   1 
                 
                 } 
               
               ⋃ 
               
                 
                   { 
                   
                     
                       
                         x 
                         2 
                       
                       + 
                       
                         a 
                         ⁢ 
                         x 
                       
                       + 
                       b 
                     
                     ∈ 
                     
                       
                         S 
                         1 
                       
                       ⁢ 
                       
                         ⁢ 
                         
                           
                             [ 
                             
                               
                                 x 
                                 2 
                               
                               + 
                               
                                 a 
                                 ⁢ 
                                 x 
                               
                               + 
                               b 
                             
                             ] 
                           
                           ⁢ 
                           
                             
 
                           
                           ⁢ 
                           are 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           equivalence 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           classes 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           distinct 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           to 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           each 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           other 
                         
                       
                     
                   
                   } 
                 
                 . 
               
             
           
         
       
     
     Generate the binary periodic sequence cluster    1  by:
         selecting one generator γ from F q * so that the elements in F q  are denoted as {γ 0 , γ 1 , . . . , γ q-2 } and select one polynomial from each equivalence class [x 2 +ax+b] of S 1 .       

     For f(x) in each T 1 , define: 
     
       
         
           
             
               
                 s 
                 f 
               
               = 
               
                 { 
                 
                   
                     η 
                     ⁡ 
                     
                       ( 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           1 
                           ) 
                         
                       
                       ) 
                     
                   
                   , 
                   
                     η 
                     ⁡ 
                     
                       ( 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           γ 
                           ) 
                         
                       
                       ) 
                     
                   
                   , 
                   … 
                   ⁢ 
                   
                       
                   
                   , 
                   
                     η 
                     ⁡ 
                     
                       ( 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           
                             γ 
                             
                               q 
                               - 
                               2 
                             
                           
                           ) 
                         
                       
                       ) 
                     
                   
                 
                 } 
               
             
             ; 
           
         
       
         
         
           
             the binary periodic sequence cluster    1  is:    1 ={s f , f(x)∈T 1 }, and furthermore, 
           
         
       
    
     
       
         
           
             
               ℱ 
               1 
             
             = 
             
               
                 { 
                 
                   
                     
                       
                         η 
                         ⁡ 
                         
                           ( 
                           
                             
                               x 
                               2 
                             
                             + 
                             ax 
                             + 
                             b 
                           
                           ) 
                         
                       
                       | 
                       x 
                     
                     = 
                     
                       γ 
                       0 
                     
                   
                   , 
                   
                     γ 
                     1 
                   
                   , 
                   … 
                   ⁢ 
                   
                       
                   
                   , 
                   
                     γ 
                     
                       q 
                       - 
                       2 
                     
                   
                   , 
                   
                     
                       [ 
                       
                         
                           x 
                           2 
                         
                         + 
                         
                           a 
                           ⁢ 
                           x 
                         
                         + 
                         b 
                       
                       ] 
                     
                     ∈ 
                     
                       S 
                       1 
                     
                   
                 
                 } 
               
               . 
             
           
         
       
     
     Parameters of this sequence are listed below:
         1. each sequence in the binary periodic sequence cluster    1  has a length of q−1;   2.    1  has a size of q; and;   3. if q≥17 and is an odd prime power, a correlation upper limit of the binary periodic sequence cluster    1  is Cor(   1 )≤6+└2√{square root over (q)}┘.       

     The estimation of correlation of such sequences depends on rational points of the curve y 2 +y=α(x 2 +ax b )(x 2 +cx+d), a, b, c, d∈F q . The correlation may also be estimated with a Hasse-Weil bound. 
     For example, a sequence  1  generated based on a multiplicative group is:
         [1011000101010001110100100000011100000010111010010010100010110 110001011011100011000001100101000110111101111011000101001100000110 001110110100011011010001010010010111010000001110000001001011100010 101000110111011010110011010010101100010101010101101110011010001110 100011111011111100110100100011110111001000100010011101111000100101 100111111011111000101110001011001110110101010101000110101001011001 101011011].  FIG. 3  is a schematic diagram showing auto-correlation values of a multiplicative group sequence  1  according to an embodiment of the present disclosure. Auto-correlation values of the sequence  1  are as shown in  FIG. 3 .       

     A sequence  2  generated based on a multiplicative group is:
         [1010110101100000111111011001111001011011101000011001101010011 101000100010000111001101101001111100001100101000001000100001001011 111110001000111010011000010101100011000011011011110001110010111110 101101010101011010111110100111000111101101100001100011010100001100 101110001000111111101001000010001000001010011000011111001011011001 110000100010001011100101011001100001011101101001111001101111110000 011010110].  FIG. 4  is a schematic diagram showing auto-correlation values of a multiplicative group sequence  2  according to an embodiment of the present disclosure. Auto-correlation values of the sequence  2  are as shown in  FIG. 4 .       

       FIG. 5  is a schematic diagram showing cross-correlation values of a multiplicative group sequence according to an embodiment of the present disclosure. Cross-correlation values of the sequences  1  and  2  generated based on multiplicative groups are as shown in  FIG. 5 . 
     Data simulation results of    1  are shown in table 1. 
     
       
         
           
               
               
               
               
               
               
             
               
                   
                 TABLE 1 
               
               
                   
                   
               
               
                   
                 Prime 
                 Sequence 
                 Sequence 
                 Correlation 
                 Balance 
               
               
                   
                 number p 
                 length 
                 number 
                 coefficient 
                 number 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
            
               
                   
                 31 
                 30 
                 30 
                 14 
                 16 
               
               
                   
                 67 
                 66 
                 66 
                 20 
                 34 
               
               
                   
                 127 
                 126 
                 126 
                 24 
                 64 
               
               
                   
                 257 
                 256 
                 256 
                 34 
                 128 
               
               
                   
                 521 
                 520 
                 520 
                 48 
                 260 
               
               
                   
                 1033 
                 1032 
                 1032 
                 66 
                 516 
               
               
                   
                   
               
            
           
         
       
     
     The above scheme considers all quadratic polynomials, but leads to the loss of the correlation coefficient. Preferably, only quadratic irreducible polynomials are considered: 
     
       
         
           
             
               S 
               1 
               ′ 
             
             = 
             
               
                 { 
                 
                   
                     
                       x 
                       2 
                     
                     + 
                     ax 
                     + 
                     b 
                   
                   , 
                   
                     a 
                     ∈ 
                     
                       F 
                       q 
                       * 
                     
                   
                   , 
                   
                     b 
                     ∈ 
                     
                       F 
                       q 
                     
                   
                   , 
                   
                     
                       x 
                       2 
                     
                     + 
                     ax 
                     + 
                     
                       b 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       is 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       an 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       irreducible 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       polynomial 
                     
                   
                 
                 } 
               
               . 
             
           
         
       
     
     A cardinality of the set S 1 ′ is equal to ½q(q−1)−½(q−1)=½(q−1) 2 . 
     Also, the equivalence class is defined according to the previous definition. 
     There are exactly q−1 elements in each equivalence class. As a result, there are exactly ½(q−1) equivalence classes. 
     Select one polynomial from each of the above equivalence classes to form a set T 1 ′, and obtain a set    1 ′={s f : f∈T 1 ′}; 
     
       
         
           
             
               T 
               1 
               ′ 
             
             = 
             
               
                 { 
                 
                   x 
                   - 
                   1 
                 
                 } 
               
               ⋃ 
               
                 
                   { 
                   
                     
                       
                         
                           x 
                           2 
                         
                         + 
                         ax 
                         + 
                         
                           b 
                           . 
                         
                       
                       ∈ 
                       
                         S 
                         1 
                         ′ 
                       
                     
                     , 
                     
                       
                         [ 
                         
                           
                             x 
                             2 
                           
                           + 
                           ax 
                           + 
                           b 
                         
                         ] 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       are 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       equivalence 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       classes 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       distinct 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       to 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       each 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       other 
                     
                   
                   } 
                 
                 . 
               
             
           
         
       
     
     Generating the sequence cluster    1 ′ includes:
         selecting one generator γ from F q *;   for f(x) in each T 1 ′, defining:       

     
       
         
           
             
               
                 s 
                 f 
               
               = 
               
                 { 
                 
                   
                     η 
                     ⁡ 
                     
                       ( 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           1 
                           ) 
                         
                       
                       ) 
                     
                   
                   , 
                   
                     η 
                     ⁡ 
                     
                       ( 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           γ 
                           ) 
                         
                       
                       ) 
                     
                   
                   , 
                   … 
                   ⁢ 
                   
                       
                   
                   , 
                   
                     η 
                     ⁡ 
                     
                       ( 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           
                             γ 
                             
                               q 
                               - 
                               2 
                             
                           
                           ) 
                         
                       
                       ) 
                     
                   
                 
                 } 
               
             
             ; 
           
         
       
         
         
           
             defining the sequence cluster as:    1 ′={s f , f(x)∈T 1 ′}. 
           
         
       
    
     Parameters of this sequence include:
         1. each sequence in the binary periodic sequence cluster    1 ′ has a length of q−1;   2.    1 ′ has a size of ½(q−−1); and   3. if q≥11 and is an odd prime power, a correlation coefficient upper limit of    1 ′ is Cor(   1 ′)≤2+└2√{square root over (q)}┘.       

     For example, a sequence  1  generated based on a multiplicative group irreducible polynomial is:
         [1101100100001000111010011110011001101000101011011011001011000 101100101010001110100111111010000011000011010001100010001000110110 001011110101100011101001100101101000001110100000110011111110100100 101101101000000011001111101000111110100101100110100011100101000010 111001001110111011100111010011110011111010000001101000111010101100 101110010110010010010101110100110011000011010001110111101100100101 011001010].  FIG. 6  is a schematic diagram showing auto-correlation values of a multiplicative group irreducible polynomial sequence  1  according to an embodiment of the disclosure. Auto-correlation values of the sequence  1  are as shown in  FIG. 6 .       

     A sequence  2  generated based on a multiplicative group irreducible polynomial is:
         [1001011000001101011101101101000100010110110111010110000011010 010110101101000110110110101000110011110010001111010100011000111111 011101011001111000111110101000010001101100110010001111110110000110 000000010010010100010011100111100101000001010011110011100100010100 100100000000110000110111111000100110011011000100001010111110001111 001101011101111110001100010101111000100111100110001010110110110001 011010110].  FIG. 7  is a schematic diagram showing auto-correlation values of a multiplicative group irreducible polynomial sequence  2  according to an embodiment of the disclosure. Auto-correlation values of the sequence  2  are as shown in  FIG. 7 .       

       FIG. 8  is a schematic diagram showing cross-correlation values of a multiplicative group irreducible polynomial sequence according to an embodiment of the disclosure. Cross-correlation values of the sequences  1  and  2  generated based on multiplicative group irreducible polynomials are as shown in  FIG. 8 . 
     Data simulation results of    1 ′ are shown in table 2. 
     
       
         
           
               
               
               
               
               
               
             
               
                   
                 TABLE 2 
               
               
                   
                   
               
               
                   
                 Prime 
                 Sequence 
                 Sequence 
                 Correlation 
                 Balance 
               
               
                   
                 number p 
                 length 
                 number 
                 coefficient 
                 number 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
            
               
                   
                 31 
                 30 
                 15 
                 10 
                 8 
               
               
                   
                 67 
                 66 
                 33 
                 18 
                 17 
               
               
                   
                 127 
                 126 
                 63 
                 22 
                 32 
               
               
                   
                 257 
                 256 
                 128 
                 32 
                 64 
               
               
                   
                 521 
                 520 
                 260 
                 44 
                 130 
               
               
                   
                 1033 
                 1032 
                 516 
                 64 
                 258 
               
               
                   
                   
               
            
           
         
       
     
     Example 2 
     Based on an additive group structure of a prime characteristic finite field, a cyclic additive group is used to construct the sequence to obtain sequences of different lengths and better correlation coefficient characteristics. In order to obtain the cyclic addition group F q , q is an odd prime. 
     Consider a monic polynomial set: 
     
       
         
           
             
               
                 S 
                 2 
               
               = 
               
                 
                   { 
                   
                     
                       
                         x 
                         2 
                       
                       + 
                       
                         a 
                         ⁢ 
                         x 
                       
                       + 
                       b 
                     
                     , 
                     a 
                     , 
                     
                       b 
                       ∈ 
                       
                         F 
                         q 
                       
                     
                   
                   } 
                 
                 ∖ 
                 
                   { 
                   
                     
                       
                         ( 
                         
                           x 
                           - 
                           a 
                         
                         ) 
                       
                       2 
                     
                     , 
                     
                       a 
                       ∈ 
                       
                         F 
                         q 
                       
                     
                   
                   } 
                 
               
             
             ; 
           
         
       
         
         
           
             the number of sets S 2  is q 2 −q=q(q−1). The equivalence class in S 2  is defined as: x 2 +a 1 x+b 1 ˜x 2 +a 2 x+b 2 , denoted as [x 2 +a 1 x+b 1 ] if and only if there exists β∈F q , a 1 =2β+a 2 , b 1 =β 2 +βa+b 2 . There are exactly q elements in each equivalence class. As a result, there are exactly q−1 equivalence classes. 
           
         
       
    
     Select one polynomial in each equivalence class, and combine the selected polynomials with x to form a set T 2 . 
     
       
         
           
             
               T 
               2 
             
             = 
             
               
                 { 
                 x 
                 } 
               
               ⋃ 
               
                 
                   { 
                   
                     
                       
                         
                           x 
                           2 
                         
                         + 
                         ax 
                         + 
                         b 
                       
                       ∈ 
                       
                         S 
                         2 
                       
                     
                     , 
                     
                       
                         [ 
                         
                           
                             x 
                             2 
                           
                           + 
                           ax 
                           + 
                           b 
                         
                         ] 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       are 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       equivalence 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       classes 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       distinct 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       to 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       each 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       other 
                     
                   
                   } 
                 
                 . 
               
             
           
         
       
     
     The method for generating the sequence cluster    2  includes:
         for the polynomial f(x) in each T 2 , defining:       

     
       
         
           
             
               
                 u 
                 f 
               
               = 
               
                 { 
                 
                   
                     η 
                     ⁡ 
                     
                       ( 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           0 
                           ) 
                         
                       
                       ) 
                     
                   
                   , 
                   
                     η 
                     ⁡ 
                     
                       ( 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           1 
                           ) 
                         
                       
                       ) 
                     
                   
                   , 
                   … 
                   ⁢ 
                   
                       
                   
                   , 
                   
                     η 
                     ⁡ 
                     
                       ( 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           
                             q 
                             - 
                             1 
                           
                           ) 
                         
                       
                       ) 
                     
                   
                 
                 } 
               
             
             ; 
           
         
       
     
     and
         defining the sequence cluster as:    2 ={u f  f(x)∈T 2 }.       

     Parameters of this sequence are listed below:
         1. each sequence in the binary periodic sequence cluster    2  has a length of q;   2. the sequence cluster    2  has a size of q; and   3. if q≥17 and is an odd prime power, a correlation coefficient upper limit of    2  is Cor(   2 )≤5+└2√{square root over (q)}┘.       

     Data simulation results of    2  are shown in table 3. 
     
       
         
           
               
               
               
               
               
               
             
               
                   
                 TABLE 3 
               
               
                   
                   
               
               
                   
                 Prime 
                 Sequence 
                 Sequence 
                 Correlation 
                 Balance 
               
               
                   
                 number p 
                 length 
                 number 
                 coefficient 
                 number 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
            
               
                   
                 31 
                 31 
                 30 
                 13 
                 30 
               
               
                   
                 67 
                 67 
                 66 
                 19 
                 66 
               
               
                   
                 127 
                 127 
                 126 
                 23 
                 126 
               
               
                   
                 257 
                 257 
                 256 
                 33 
                 256 
               
               
                   
                 521 
                 521 
                 520 
                 47 
                 520 
               
               
                   
                 1033 
                 1033 
                 1032 
                 65 
                 1032 
               
               
                   
                   
               
            
           
         
       
     
     The above scheme only considers all quadratic polynomials, but leads to the loss of the correlation coefficient. In a preferred embodiment, quadratic irreducible polynomials in S t  are considered to form a set denoted as S 2 ′. 
     
       
         
           
             
               S 
               2 
               ′ 
             
             = 
             
               
                 { 
                 
                   
                     
                       x 
                       2 
                     
                     + 
                     
                       a 
                       ⁢ 
                       x 
                     
                     + 
                     b 
                   
                   , 
                   a 
                   , 
                   
                     b 
                     ∈ 
                     
                       F 
                       q 
                     
                   
                   , 
                   
                     
                       x 
                       2 
                     
                     + 
                     ax 
                     + 
                     
                       b 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       is 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       an 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       irreducible 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       polynomial 
                     
                   
                 
                 } 
               
               . 
             
           
         
       
     
     Select one polynomial in each equivalence class of S 2 ′, and combine the selected polynomials with x to form a set T 2 ′: 
     
       
         
           
             
               T 
               2 
               ′ 
             
             = 
             
               
                 { 
                 x 
                 } 
               
               ⋃ 
               
                 { 
                 
                   
                     
                       
                         x 
                         2 
                       
                       + 
                       ax 
                       + 
                       b 
                     
                     ∈ 
                     
                       S 
                       2 
                       ′ 
                     
                   
                   , 
                   
                     
                       [ 
                       
                         
                           x 
                           2 
                         
                         + 
                         ax 
                         + 
                         b 
                       
                       ] 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     are 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     equivalence 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     classes 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     distinct 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     to 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     each 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     other 
                   
                 
                 } 
               
             
           
         
       
     
     The method for generating the sequence cluster    2 ′ includes:
         for the polynomial f(x) in each T 2 ′, defining:       

     
       
         
           
             
               
                 u 
                 f 
               
               = 
               
                 { 
                 
                   
                     η 
                     ⁡ 
                     
                       ( 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           0 
                           ) 
                         
                       
                       ) 
                     
                   
                   , 
                   
                     η 
                     ⁡ 
                     
                       ( 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           1 
                           ) 
                         
                       
                       ) 
                     
                   
                   , 
                   … 
                   ⁢ 
                   
                       
                   
                   , 
                   
                     η 
                     ⁡ 
                     
                       ( 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           
                             q 
                             - 
                             1 
                           
                           ) 
                         
                       
                       ) 
                     
                   
                 
                 } 
               
             
             ; 
           
         
       
     
     and
         defining the sequence cluster as:       

     
       
         
           
             
               ℱ 
               2 
               ′ 
             
             = 
             
               
                 { 
                 
                   
                     u 
                     f 
                   
                   , 
                   
                     
                       f 
                       ⁡ 
                       
                         ( 
                         x 
                         ) 
                       
                     
                     ∈ 
                     
                       T 
                       2 
                       ′ 
                     
                   
                 
                 } 
               
               . 
             
           
         
       
     
     Parameters of this sequence are listed below:
         1. each sequence in the binary periodic sequence cluster    2 ′ has a length of q;   2. the sequence cluster  2 has a size of (q−1)/2; and   3. if q≥7 and is an odd prime, a correlation coefficient upper limit of    2 ′ is Cor(   4 )≤1+└2√{square root over (q)}┘.       

     For example, a sequence  1  generated based on an additive group irreducible polynomial is:
         [0010001011000011110001000000101001100000010011101101100110110 101101101111110111111000011111001011110100100100001110001000101111 100111111110011111010001000111000010010010111101001111100001111110 111111011011010110110011011011100100000011001010000001000111100001 101000100100010100100111000111111101000100010101110110100000001100 001111010001011110000110000000101101110101000100010111111100011100 1001010001].  FIG. 9  is a schematic diagram showing auto-correlation values of an additive group irreducible polynomial sequence  1  according to an embodiment of the disclosure. Auto-correlation values of the sequence  1  generated based on an additive group irreducible polynomial are as shown in  FIG. 9 .       

     A sequence  2  generated based on an additive group irreducible polynomial is:
         [1010010000111010100001011001000100010111100001110011010111000 001100101110111101110101001110010100001001010011011111111001011000 011110110111100001101001111111101100101001000010100111001010111011 110111010011000001110101100111000011110100010001001101000010101110 000100101000100100110101110010101000001001100010001111001011101100 101011111011111010100110111010011110001000110010000010101001110101 1001001000].  FIG. 10  is a schematic diagram showing auto-correlation values of an additive group irreducible polynomial sequence  2  according to an embodiment of the disclosure. Auto-correlation values of the sequence  2  generated based on an additive group irreducible polynomial are as shown in  FIG. 10 .       

       FIG. 11  is a schematic diagram showing cross-correlation values of an additive group irreducible polynomial sequence according to an embodiment of the disclosure. Cross-correlation values of the sequences  1  and  2  generated based on additive group irreducible polynomial sequences are as shown in  FIG. 11 . 
     Data simulation results of    2 ′ are shown in table 4. 
     
       
         
           
               
               
               
               
               
               
             
               
                   
                 TABLE 4 
               
               
                   
                   
               
               
                   
                 Prime 
                 Sequence 
                 Sequence 
                 Correlation 
                 Balance 
               
               
                   
                 number p 
                 length 
                 number 
                 coefficient 
                 number 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
            
               
                   
                 31 
                 31 
                 15 
                 9 
                 15 
               
               
                   
                 67 
                 67 
                 33 
                 17 
                 33 
               
               
                   
                 127 
                 127 
                 63 
                 21 
                 63 
               
               
                   
                 257 
                 257 
                 128 
                 31 
                 128 
               
               
                   
                 521 
                 521 
                 260 
                 43 
                 260 
               
               
                   
                 1033 
                 1033 
                 516 
                 63 
                 516 
               
               
                   
                   
               
            
           
         
       
     
     Example 3 
     In a positioning signal generation environment in 5G, firstly, since the PRSID of LTE ranges from 0 to 4095, a slot number n s  in each frame ranges from 0 to 20, and the number 1 of each OFDM symbol ranges from 0 to 6, there are 4096*20*7=573440 different combinations of these three parameters. Taking the closest prime q=573451 (closest to 573440) as the initial parameter, and generating a multiplicative group F 573451 * is generated. 
     For selection of the positioning sequence in 5G, a finite field F 573451  for q=573451 is selected as the finite field used for generating the positioning sequence.  FIG. 12  is a schematic diagram showing auto-correlation values of a positioning reference signal generated according to an embodiment of the present disclosure. Auto-correlation values of the generated positioning reference signal are as shown in  FIG. 12 .  FIG. 13  is a schematic diagram showing cross-correlation values of two positioning reference signals generated according to an embodiment strip the present disclosure. Cross-correlation values of the two positioning reference signals are as shown in  FIG. 13 . 
     A method for truncating a 5G positioning sequence includes taking a truncated code with a length of 400, from bit 1600 to bit 2000 of the positioning sequence, as the positioning code, which is consistent with the 3GPP standard. 
     Secondly, according to the above sequence construction method, each polynomial f(x)=x 2 +ax+b may determine a sequence cluster, in which only parameters a and b are included. Three parameters are mapped to LTE according to different combinations of a and b, which specifically includes: 
     
       
         
           
             { 
             
               
                 
                   
                     
                       a 
                       = 
                       1 
                     
                   
                 
                 
                   
                     
                       b 
                       = 
                       
                         
                           180 
                           · 
                           
                             N 
                             ID 
                             PRS 
                           
                         
                         + 
                         
                           100 
                           · 
                           
                             ⌊ 
                             
                               
                                 n 
                                 s 
                               
                               / 
                               10 
                             
                             ⌋ 
                           
                         
                         + 
                         
                           10 
                           · 
                           1 
                         
                         + 
                         
                           ( 
                           
                             
                               n 
                               s 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             mod 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             10 
                           
                           ) 
                         
                       
                     
                   
                 
               
               , 
             
           
         
       
     
     where N ID   PRS  the positioning reference signal (PRS) ID, 1 is an OFDM symbol number in a slot, and n s  is a slot number within a frame. 
     According to the 3GPP standard, different C init  are randomly generated to generate the Gold sequence, and 400 bits after bit 1600 are truncated. The same random parameters and the same truncation method are used to generate the sequence in the embodiment of the present disclosure and a Kasami sequence. Parts of the three sequences are selected for comparison. The comparison results are shown in table 5. 
     
       
         
           
               
               
               
               
             
               
                 TABLE 5 
               
               
                   
               
               
                   
                   
                   
                 Binary periodic 
               
               
                   
                   
                   
                 sequence in 
               
               
                   
                 Gold  
                   
                 embodiments of the 
               
               
                   
                 sequence 
                 Kasami 
                 present disclosure 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
            
               
                 Average 
                 59.55 
                 59.52 
                 59.18 
               
               
                 auto-correlation 
                   
                   
                   
               
               
                 Maximum 
                 104 
                 104 
                 100 
               
               
                 auto-correlation 
                   
                   
                   
               
               
                 Average 
                 63.384386 
                 63.351228 
                 63.296617 
               
               
                 cross-correlation 
                   
                   
                   
               
               
                 Maximum 
                 126 
                 118 
                 116 
               
               
                 cross-correlation 
               
               
                   
               
            
           
         
       
     
     The positioning process is described below. 
     After the transmitting end generates a positioning code, the positioning code is modulated according to the 3GPP signal modulation scheme to generate a positioning signal. Then, the positioning signal is mapped to a transmitting antenna port for transmission according to the 3GPP resource grid mapping scheme. After receiving the signal, the receiving end performs cross-correlation to obtain an arrival time of the signal and thus a distance to the signal transmitting end, so as to select an appropriate algorithm for positioning. 
     Through the description of the above implementations, those skilled in the art can clearly understand that the method according to the above embodiment may be implemented by means of software plus a necessary general hardware platform. Obviously, it may also be implemented by hardware, but in most cases, the former is preferred. Based on such understanding, the technical solutions of the present invention essentially or, in other words, a part thereof contributing to the prior art, can be embodied in a form of a software product, wherein the software product is stored in a storage medium (such as an ROM/RAM, a disk, or an optical disc) and includes a number of instructions to make a terminal device (which may be a mobile phone, a computer, a server, or a network device, etc.) to execute the methods of the various embodiments of the present disclosure. 
     Embodiment 2 
     This embodiment further provides a positioning and transmission apparatus based on a binary periodic sequence, which is configured to implement the above embodiments and preferred implementations. Details which have been explained will not be repeated here. As used herein, the term “module” may be a combination of software and/or hardware that can realize a preset function. The apparatus described in the following embodiments is preferably implemented in software, but hardware, or a combination of software and hardware, is also possible and contemplated. 
       FIG. 14  is a block diagram of a positioning apparatus based on a binary periodic sequence according to an embodiment of the present disclosure. As shown in  FIG. 14 , the positioning apparatus includes: 
     a first selection module  142  configured to select one polynomial from each equivalence class of a quadratic polynomial set S to determine a set T; 
     a construction module  144  configured to construct a binary periodic sequence cluster according to the set T; 
     a generating module  146  configured to generate a positioning signal according to the binary periodic sequence cluster; and 
     a positioning processing module  148  configured to perform positioning processing according to the positioning signal. 
     Optionally, the generating module  146  includes: 
     a selecting unit configured to select a positioning sequence from the binary periodic sequence cluster according to a preset parameter; 
     a determining unit configured to take a truncated sequence with a preset length from the positioning sequence and determining the truncated sequence as a positioning reference sequence; and a generating unit configured to generate the positioning signal by modulating the positioning reference sequence. 
     Optionally, the positioning processing module  148  includes: 
     a transmitting unit configured to map the positioning signal to an antenna port at a transmitting end for transmission. The positioning signal is configured to instruct a receiving end, after receiving the positioning signal, to: perform cross-correlation to obtain an arrival time of the positioning signal, determine a distance from a transmitting end according to the arrival time, and perform positioning according to the distance. 
     The transmitting end may be the positioning apparatus itself, or may include the positioning apparatus or be independent of the positioning device. The receiving end may form a positioning system with the positioning device. 
     Optionally, the positioning apparatus further includes: 
     a second selection module configured to select one generator γ from F q *, where F q  is a finite field containing q elements, q is an odd prime or an odd prime power, and F q * is a set of all nonzero elements in F q . 
     Optionally, the first selection module  142  is further configured to select one polynomial f(x) from each equivalence class of a quadratic polynomial set S 1  and combine the polynomial f(x) with x−1 to form a set T 1  by: 
     
       
         
           
             
               
                 T 
                 1 
               
               = 
               
                 
                   { 
                   
                     x 
                     - 
                     1 
                   
                   } 
                 
                 ⋃ 
                 
                   { 
                   
                     
                       
                         
                           x 
                           2 
                         
                         + 
                         ax 
                         + 
                         b 
                       
                       ∈ 
                       
                         S 
                         1 
                       
                     
                     , 
                     
                       
                         [ 
                         
                           
                             x 
                             2 
                           
                           + 
                           ax 
                           + 
                           b 
                         
                         ] 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       are 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       equivalence 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       classes 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       distinct 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       to 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       each 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       other 
                     
                   
                   } 
                 
               
             
             , 
             where 
           
         
       
       
         
           
             
               S 
               1 
             
             = 
             
               
                 { 
                 
                   
                     
                       x 
                       2 
                     
                     + 
                     ax 
                     + 
                     b 
                   
                   , 
                   
                     a 
                     ∈ 
                     
                       F 
                       q 
                       * 
                     
                   
                   , 
                   
                     b 
                     ∈ 
                     
                       F 
                       q 
                     
                   
                 
                 } 
               
               ⁢ 
               \ 
               ⁢ 
               
                 
                   { 
                   
                     
                       
                         ( 
                         
                           x 
                           - 
                           a 
                         
                         ) 
                       
                       2 
                     
                     , 
                     
                       a 
                       ∈ 
                       
                         F 
                         q 
                         * 
                       
                     
                   
                   } 
                 
                 . 
               
             
           
         
       
     
     Optionally, the construction module  144  is further configured to construct a binary periodic sequence cluster    1  from the generator γ and the set T 1 : 
     
       
         
           
             
               F_ 
               ⁢ 
               1 
             
             = 
             
               { 
               
                 s_f 
                 , 
                 
                   
                     
                       ℱ 
                       1 
                     
                     = 
                     
                       { 
                       
                         
                           s 
                           f 
                         
                         , 
                         
                           
                             f 
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                           ∈ 
                           
                             T 
                             1 
                           
                         
                       
                       } 
                     
                   
                   ; 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       s 
                       f 
                     
                     = 
                     
                       { 
                       
                         
                           η 
                           ⁡ 
                           
                             ( 
                             
                               f 
                               ⁡ 
                               
                                 ( 
                                 1 
                                 ) 
                               
                             
                             ) 
                           
                         
                         , 
                         
                           η 
                           ⁡ 
                           
                             ( 
                             
                               f 
                               ⁡ 
                               
                                 ( 
                                 γ 
                                 ) 
                               
                             
                             ) 
                           
                         
                         , 
                         
                           . 
                           
                               
                           
                           . 
                         
                         ⁢ 
                         
                             
                         
                         , 
                         
                           η 
                           ⁡ 
                           
                             ( 
                             
                               f 
                               ⁡ 
                               
                                 ( 
                                 
                                   γ 
                                   
                                     q 
                                     - 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             ) 
                           
                         
                       
                       } 
                     
                   
                   ; 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       η 
                       ⁡ 
                       
                         ( 
                         a 
                         ) 
                       
                     
                     = 
                     
                       { 
                       
                         
                           
                             
                               1 
                               , 
                             
                           
                           
                             
                               if 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               a 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               is 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               a 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               nonzero 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               square 
                             
                           
                         
                         
                           
                             
                               
                                 - 
                                 1 
                               
                               , 
                             
                           
                           
                             
                               
                                 if 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 a 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 is 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 a 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 non 
                               
                               - 
                               square 
                             
                           
                         
                         
                           
                             
                               0 
                               , 
                             
                           
                           
                             
                               
                                 if 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 a 
                               
                               = 
                               0 
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
         
         
           
             where η is a quadratic multiplicative character from F q  to C*, C is a complex number set excluding 0, and α is any element in F q *. 
           
         
       
    
     Optionally, each sequence in the binary periodic sequence cluster    1  has a length of q−1; 
     the binary periodic sequence cluster    1  has a size of q; and 
     if q≥17 and is an odd prime power, a correlation upper limit of the binary periodic sequence cluster    1  is Cor(   1 )≤6+└2√{square root over (q)}┘. 
     Optionally, the first selection module  142  is further configured to, in the case wherex 2 +ax+b in S 1  is an irreducible polynomial, 
     
       
         
           
             
               
                 S 
                 1 
                 ′ 
               
               = 
               
                 { 
                 
                   
                     
                       x 
                       2 
                     
                     + 
                     ax 
                     + 
                     b 
                   
                   , 
                   
                     a 
                     ∈ 
                     
                       F 
                       q 
                       * 
                     
                   
                   , 
                   
                     b 
                     ∈ 
                     
                       F 
                       q 
                     
                   
                   , 
                   
                     
                       x 
                       2 
                     
                     + 
                     ax 
                     + 
                     
                       b 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       is 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       an 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       irreducible 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       polynomial 
                     
                   
                 
                 } 
               
             
             ; 
           
         
       
     
     select one polynomial f(x) from each equivalence class of a quadratic polynomial set S 1 ′ and combining the polynomial f(x) with x−1 to form a set T 1 ′: 
     
       
         
           
             
               T 
               1 
               ′ 
             
             = 
             
               
                 { 
                 
                   x 
                   - 
                   1 
                 
                 } 
               
               ⋃ 
               
                 
                   { 
                   
                     
                       
                         
                           x 
                           2 
                         
                         + 
                         
                           a 
                           ⁢ 
                           x 
                         
                         + 
                         
                           b 
                           . 
                         
                       
                       ∈ 
                       
                         S 
                         1 
                         ′ 
                       
                     
                     , 
                     
                       
                         [ 
                         
                           
                             x 
                             2 
                           
                           + 
                           
                             a 
                             ⁢ 
                             x 
                           
                           + 
                           b 
                         
                         ] 
                       
                       ⁢ 
                       
                         
 
                       
                       ⁢ 
                       are 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       equivalence 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       classes 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       distinct 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       to 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       each 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       other 
                     
                   
                   } 
                 
                 . 
               
             
           
         
       
     
     Optionally, the construction module  144  is further configured to construct a binary periodic sequence cluster    1 ′ from the generator γ and the set T 1 ′: 
     
       
         
           
             
               
                 ℱ 
                 1 
                 ′ 
               
               = 
               
                 { 
                 
                   
                     s 
                     f 
                   
                   , 
                   
                     
                       f 
                       ⁡ 
                       
                         ( 
                         x 
                         ) 
                       
                     
                     ∈ 
                     
                       T 
                       1 
                       ′ 
                     
                   
                 
                 } 
               
             
             ; 
           
         
       
       
         
           
             
               
                 s 
                 f 
               
               = 
               
                 { 
                 
                   
                     η 
                     ⁡ 
                     
                       ( 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           1 
                           ) 
                         
                       
                       ) 
                     
                   
                   , 
                   
                     η 
                     ⁡ 
                     
                       ( 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           γ 
                           ) 
                         
                       
                       ) 
                     
                   
                   , 
                   
                     . 
                     
                         
                     
                     . 
                   
                   ⁢ 
                   
                       
                   
                   , 
                   
                     η 
                     ⁡ 
                     
                       ( 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           
                             γ 
                             
                               q 
                               - 
                               2 
                             
                           
                           ) 
                         
                       
                       ) 
                     
                   
                 
                 } 
               
             
             ; 
           
         
       
       
         
           
             
               η 
               ⁡ 
               
                 ( 
                 a 
                 ) 
               
             
             = 
             
               { 
               
                 
                   
                     
                       1 
                       , 
                     
                   
                   
                     
                       if 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       a 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       is 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       a 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       nonzero 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       square 
                     
                   
                 
                 
                   
                     
                       
                         - 
                         1 
                       
                       , 
                     
                   
                   
                     
                       
                         if 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         a 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         is 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         a 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         non 
                       
                       - 
                       square 
                     
                   
                 
                 
                   
                     
                       0 
                       , 
                     
                   
                   
                     
                       
                         if 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         a 
                       
                       = 
                       0 
                     
                   
                 
               
             
           
         
       
     
     where η is a quadratic multiplicative character from F q  to C*, C is a complex number set excluding 0, and α is any element in F q *. 
     Optionally, each sequence in the binary periodic sequence cluster    1 ′ has a length of q−1; 
     the binary periodic sequence cluster    1 ′ has a size of ½ (q−1); and 
     if q≥11 and is an odd prime power, a correlation coefficient upper limit of the binary periodic sequence cluster    1 ′ is Cor(   1 ′)≤2+└2√{square root over (q)}┘. 
     Optionally, the positioning apparatus further includes: 
     a first dividing module configured to perform equivalence class division on the polynomials in the quadratic polynomial set S, and determine, if and only if there exists β∈F q * such that a 1 =βa 2 ,b1=β 2 b 2 , an equivalence relation between x 2 +a i x+b 1  and x 2 +a 2 x+b 2 , and determining an equivalence class of the quadratic polynomial set S to be [x 2 +a i x+b 1 ]. 
     Optionally, the first selection module  142  is further configured to select one polynomial f(x) from each equivalence class of a quadratic polynomial set S 2  and combine the polynomial f(x) with x to form a set T 2 : 
     
       
         
           
             
               
                 T 
                 2 
               
               = 
               
                 
                   { 
                   x 
                   } 
                 
                 ⋃ 
                 
                   { 
                   
                     
                       
                         
                           x 
                           2 
                         
                         + 
                         
                           a 
                           ⁢ 
                           x 
                         
                         + 
                         b 
                       
                       ∈ 
                       
                         S 
                         2 
                       
                     
                     , 
                     
                       
                         [ 
                         
                           
                             x 
                             2 
                           
                           + 
                           
                             a 
                             ⁢ 
                             x 
                           
                           + 
                           b 
                         
                         ] 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       are 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       equivalence 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       classes 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       distinct 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       to 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       each 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       other 
                     
                   
                   } 
                 
               
             
             ; 
           
         
       
     
     S 2 ={x 2  ax+b, a,b∈F q }\{(x−a) 2 , a∈F q }, where F q  is a finite field containing q elements, and q is an odd prime. 
     Optionally, the construction module  144  is further configured to construct a binary periodic sequence cluster    2  from the set T 2 : 
     
       
         
           
             
               
                 ℱ 
                 2 
               
               = 
               
                 { 
                 
                   
                     u 
                     f 
                   
                   , 
                   
                     
                       f 
                       ⁡ 
                       
                         ( 
                         x 
                         ) 
                       
                     
                     ∈ 
                     
                       T 
                       2 
                     
                   
                 
                 } 
               
             
             ; 
           
         
       
       
         
           
             
               
                 u 
                 f 
               
               = 
               
                 { 
                 
                   
                     η 
                     ⁡ 
                     
                       ( 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           0 
                           ) 
                         
                       
                       ) 
                     
                   
                   , 
                   
                     η 
                     ⁡ 
                     
                       ( 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           1 
                           ) 
                         
                       
                       ) 
                     
                   
                   , 
                   … 
                   ⁢ 
                   
                       
                   
                   , 
                   
                     η 
                     ⁡ 
                     
                       ( 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           
                             q 
                             - 
                             1 
                           
                           ) 
                         
                       
                       ) 
                     
                   
                 
                 } 
               
             
             ; 
           
         
       
       
         
           
             
               η 
               ⁡ 
               
                 ( 
                 a 
                 ) 
               
             
             = 
             
               { 
               
                 
                   
                     
                       1 
                       , 
                     
                   
                   
                     
                       if 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       a 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       is 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       a 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       nonzero 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       square 
                     
                   
                 
                 
                   
                     
                       
                         - 
                         1 
                       
                       , 
                     
                   
                   
                     
                       
                         if 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         a 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         is 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         a 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         non 
                       
                       - 
                       square 
                     
                   
                 
                 
                   
                     
                       0 
                       , 
                     
                   
                   
                     
                       
                         if 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         a 
                       
                       = 
                       0 
                     
                   
                 
               
             
           
         
       
         
         
           
             where η is a quadratic multiplicative character from F q  to C*, C is a complex number set excluding 0, and a is any element in F q *. 
           
         
       
    
     Optionally, each sequence in the binary periodic sequence cluster    2  has a length of q; 
     the binary periodic sequence cluster    2  has a size of q; and 
     if q≥17 and is an odd prime, a correlation coefficient upper limit of the binary periodic sequence cluster    2  is Cor(   2 )≤5+└2√{square root over (q)}┘. 
     Optionally, the first selection module  142  is further configured to, in the case where the x 2 +ax+b in S 2  is an irreducible polynomial, 
     
       
         
           
             
               
                 S 
                 2 
                 ′ 
               
               = 
               
                 { 
                 
                   
                     
                       x 
                       2 
                     
                     + 
                     
                       a 
                       ⁢ 
                       x 
                     
                     + 
                     b 
                   
                   , 
                   a 
                   , 
                   
                     b 
                     ∈ 
                     
                       F 
                       q 
                     
                   
                   , 
                   
                     
                       x 
                       2 
                     
                     + 
                     
                       a 
                       ⁢ 
                       x 
                     
                     + 
                     
                       b 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       is 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       an 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       irreducible 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       polynomial 
                     
                   
                 
                 ⁢ 
                 
                     
                 
                 } 
               
             
             ; 
           
         
       
     
     select one polynomial f(x) from each equivalence class of a quadratic polynomial set S 2 ′ and combine the polynomial f(x) with x to form a set T 2 ′: 
     
       
         
           
             
               T 
               2 
               ′ 
             
             = 
             
               
                 { 
                 x 
                 } 
               
               ⋃ 
               
                 
                   { 
                   
                     
                       
                         
                           x 
                           2 
                         
                         + 
                         
                           a 
                           ⁢ 
                           x 
                         
                         + 
                         b 
                       
                       ∈ 
                       
                         S 
                         2 
                         ′ 
                       
                     
                     , 
                     
                       
                         [ 
                         
                           
                             x 
                             2 
                           
                           + 
                           
                             a 
                             ⁢ 
                             x 
                           
                           + 
                           b 
                         
                         ] 
                       
                       ⁢ 
                       
                         
 
                       
                       ⁢ 
                       are 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       equivalence 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       classes 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       distinct 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       to 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       each 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       other 
                     
                   
                   } 
                 
                 . 
               
             
           
         
       
     
     Optionally, the construction module is further configured to a binary periodic sequence cluster    2 ′ from the set T 2 : 
     
       
         
           
             
               
                 ℱ 
                 2 
                 ′ 
               
               = 
               
                 { 
                 
                   
                     u 
                     f 
                   
                   , 
                   
                     
                       f 
                       ⁡ 
                       
                         ( 
                         x 
                         ) 
                       
                     
                     ∈ 
                     
                       T 
                       2 
                       ′ 
                     
                   
                 
                 } 
               
             
             ; 
           
         
       
       
         
           
             
               
                 u 
                 f 
               
               = 
               
                 { 
                 
                   
                     η 
                     ⁡ 
                     
                       ( 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           0 
                           ) 
                         
                       
                       ) 
                     
                   
                   , 
                   
                     η 
                     ⁡ 
                     
                       ( 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           γ 
                           ) 
                         
                       
                       ) 
                     
                   
                   , 
                   
                     . 
                     
                         
                     
                     . 
                   
                   ⁢ 
                   
                       
                   
                   , 
                   
                     η 
                     ⁡ 
                     
                       ( 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           
                             q 
                             - 
                             1 
                           
                           ) 
                         
                       
                       ) 
                     
                   
                 
                 } 
               
             
             ; 
           
         
       
       
         
           
             
               η 
               ⁡ 
               
                 ( 
                 a 
                 ) 
               
             
             = 
             
               { 
               
                 
                   
                     
                       1 
                       , 
                     
                   
                   
                     
                       if 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       a 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       is 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       a 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       nonzero 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       square 
                     
                   
                 
                 
                   
                     
                       
                         - 
                         1 
                       
                       , 
                     
                   
                   
                     
                       
                         if 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         a 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         is 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         a 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         non 
                       
                       - 
                       square 
                     
                   
                 
                 
                   
                     
                       0 
                       , 
                     
                   
                   
                     
                       
                         if 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         a 
                       
                       = 
                       0 
                     
                   
                 
               
             
           
         
       
         
         
           
             where η is a quadratic multiplicative character from F q  to C*, C is a complex number set excluding 0, and a is any element in F q *. 
           
         
       
    
     Optionally, each sequence in the binary periodic sequence cluster    2 ′ has a length of q; 
     the binary periodic sequence cluster    2 ′ has a size of (q−1)/2; and 
     if q≥7 and is an odd prime, a correlation coefficient upper limit of the binary periodic sequence cluster    2 ′ is Cor(   4 )≤1+└2√{square root over (q)}┘. 
     Optionally, the positioning apparatus further includes: 
     a second dividing module configured to perform before the first selection module  142  selects the polynomial from each equivalence class of a quadratic polynomial set S to determine a set T, equivalence class division on the polynomials in the quadratic polynomial set S. The division operation includes: 
     the second dividing module determining, if and only if there exists β∈F q , a 1 =2β+a 2 , b 1 =β 2 +βa+b 2 , an equivalence relation between x 2 +a 1 x+b 1  and x 2 +a 2 x+b 2 , and determining an equivalence class of the quadratic polynomial set S to be [x 2 +a 1 x+b 1 ]. 
     It should be noted that each of the above modules may be implemented by software or hardware. For the latter, it may be implemented by, but are not limited to: the above modules all located in the same processor; or the above modules each located in different processors in any combination. 
     Embodiment 3 
     An embodiment of the disclosure further provides a storage medium having a computer program stored thereon, which computer program is configured to be executed to cause steps of any one of the above method embodiments to be implemented. 
     Optionally, in this embodiment, the storage medium may be configured to store a computer program for implementing the steps of: 
     S 11 , selecting one polynomial from each equivalence class of a quadratic polynomial set S to determine a set T; 
     S 12 , constructing a binary periodic sequence cluster according to the set T; S 13 , generating a positioning signal according to the binary periodic sequence cluster; and 
     S 14 , performing positioning processing according to the positioning signal. 
     Optionally, in this embodiment, the storage medium may include, but is not limited to: a U disk, a read-only memory (ROM), a random access memory (RAM), a mobile hard disk, a disk or optical disk, and other media that can store a computer program. 
     Embodiment 4 
     An embodiment of the disclosure further provides an electronic apparatus, including a memory having a computer program stored thereon and a processor configured to execute the computer program to perform steps of any one of the above method embodiments. 
     Optionally, the electronic apparatus may further include a transmission device and an input/output device. The transmission device is coupled to the processor, and the input/output device is coupled to the processor. 
     Optionally, in this embodiment, the processor may be configured to execute the following steps via the computer program: 
     S 11 , selecting one polynomial from each equivalence class of a quadratic polynomial set S to determine a set T; 
     S 12 , constructing a binary periodic sequence cluster according to the set T; S 13 , generating a positioning signal according to the binary periodic sequence cluster; and 
     S 14 , performing positioning processing according to the positioning signal. 
     Optionally, specific examples in the present embodiment may refer to the examples described in the foregoing embodiments and alternative implementations, which will not be repeated in the present embodiment. 
     Obviously, a person skilled in the art would understand that the above modules and steps of the present disclosure can be realized by using a universal computing device, can be integrated in a single computing device or distributed on a network that consists of a plurality of computing devices; and alternatively, they can be realized by using the executable program code of the computing device, so that they can be stored in a storage device and executed by the computing device, in some cases, can perform the shown or described steps in a sequence other than herein, or they are made into various integrated circuit modules respectively, or a plurality of modules or steps thereof are made into a single integrated circuit module, thus to be realized. In this way, the present disclosure is not restricted to any particular hardware and software combination. 
     The descriptions above are only preferred embodiments of the present disclosure, which are not used to restrict the present disclosure. For those skilled in the art, the present disclosure may have various changes and variations. Any amendments, equivalent substitutions, improvements, etc. within the principle of the disclosure are all included in the scope of the protection defined by the appended claims of the disclosure.