# Patent

## Title
Method of transmitting information between a plurality of radioelectric stations and associated transmission network

## Abstract

A method of transmitting information between a plurality of radioelectric stations and an associated transmission network are disclosed. In one aspect, the method transmits information between radioelectric stations, each station including a transmitter and a receiver, the information including NbMot words of data, NbMot being an integer >1. The transmission method includes: determining at least one polynomial of degree NbMot−1, each of the NbMot coefficients of the polynomial corresponding to a respective word, the polynomial having an indeterminacy, calculating NbRepet polynomial values for the polynomial, NbRepet being an integer >1, each polynomial value being calculated for a respective predetermined value of the indeterminacy, transmitting the calculated polynomial values from one radioelectric station to at least one other radioelectric station, receiving polynomial values by the other radioelectric station, and determining, via the other radioelectric station, the NbMot words from a Lagrange interpolation of the received polynomial values.

## Background
INCORPORATION BY REFERENCE TO ANY PRIORITY APPLICATIONS
This application is a continuation of Application No. PCT/EP2013/069530, filed Sep. 19, 2013, which claims the benefit under 35 U.S.C. §119 of French Application No. 12 02521, filed Sep. 21, 2012, which are herein incorporated by reference in their entirety.


BACKGROUND
Field
The described technology generally relates to a method for transmitting information between a plurality of radioelectric stations, each station comprising a transmitter and a receiver, the information comprising NbMot words of data, NbMot being an integer strictly greater than 1. The described technology also relates to a transmission network comprising a plurality of radioelectric stations.
Description of the Related Technology
The described technology in particular relates to the case where the information to be transmitted has a size larger than the size of a radio packet dedicated to transporting that information. The described technology also relates to the case where it is necessary to repeat the sending of the information several times in order to guarantee, with a high likelihood, the minimum likelihood depending on the desired quality of service and for example equal to 0.99, the correct reception of at least one copy of said information.
The described technology in particular applies to a transmission network working by frequency hopping in a predetermined frequency plane, the frequency used for the data transmission being kept during a plateau, called a frequency hopping (FH) plateau, and changing from one FH plateau to another according to a frequency change law.
A transmission method of the aforementioned type is standard. The information to be transmitted is conveyed in a frame comprising NbMot data words T1, . . . , TNbMot, and defined as follows:
[T1∥T2∥ . . . ∥TNbMot], where ∥ symbolizes a data concatenation.
The transmission of the frame is repeated several times in order to guarantee, with a high likelihood, the correct reception of at least one copy of the information. The transmission method comprises a synchronization mechanism between frames, to allow the receiving station to identify each transmitted frame correctly.
Due to the repetition, one of the NbMot words of the frame, such as the first word T1, also contains a frame indicator TPS, said indicator TPS evolving over the course of the repetitions and making it possible to number the frames successively transmitted.
A synchronization information then comprises the synchronization mechanism so that the receiving station recognizes the division into frames and the frame indicator TPS. Upon each repetition of the frame, the value, denoted TPS(i), of the frame indicator TPS evolves such that the receiving station globally knows how to position the received packet.
The transmitting station then transmits the data words T1, . . . , TNbMot in different packets as follows:

    
    
        packet number 1 containing the first word T1 with the frame indicator with value TPS(1),
        packet number 2 containing the second word T2, . . . ,
        packet number NbMot containing the word TNbMot, the packets numbered 1 to NbMot forming the first frame,
        packet number NbMot+1 containing the first word T1 with the frame indicator with value TPS(2),
        packet number NbMot+2 containing the second word T2, . . . ,
        packet number 2×NbMot containing the word TNbMot, the packets numbered NbMot+1 to 2×NbMot forming the second frame,
        packet number 2×NbMot+1 containing the first word T1 with the frame indicator with value TPS(3),
        packet number 2×NbMot+2 containing the second word T2, . . . , and so forth up to a maximum number of frames, defined as a function of a likelihood that the receiving station will correctly obtain each of the data words T1, . . . , TNbMot, that likelihood also being called quality of service.
    
    


The word T1 with the frame indicator whose value TPS(i) evolves is repeated via packets having a number adjacent to 1 modulo NbMot. Likewise, the word T2 is sent in packets having a number adjacent to 2 modulo NbMot, and so forth. The word TNbMot is repeated in packets having a number adjacent to 0 modulo NbMot.
Consequently, for the receiving station to obtain a copy of each word T1, . . . , TNbMot, it must be able to correctly decode NbMot packets not necessarily belonging to the same frame, but each having a different number considering it modulo NbMot.
However, with such a transmission method, the maximum number of frames transmitted in order to guarantee the desired quality of service is high, and the transmission network is then particularly slow to transmit information.
SUMMARY OF CERTAIN INVENTIVE ASPECTS
One inventive aspect is a transmission method and network making it possible to reduce the quantity of data transmitted by the transmitting station to the receiving station(s), to convey said information comprising the data words T1, . . . , TNbMot, while offering the same quality of service.
Another aspect is a transmission method of the aforementioned type, wherein the method comprises the following steps:

    
    
        determining at least one polynomial of degree NbMot−1, each of the NbMot coefficients of the polynomial corresponding to a respective word, each polynomial having a variable,
        calculating NbRepet polynomial values for each polynomial, NbRepet being an integer strictly greater than 1, each polynomial value being calculated for a respective predetermined value of the variable,
        transmitting the calculated polynomial values from one radioelectric station to at least one other radioelectric station,
        receiving polynomial values by each other radioelectric station, and
        determining, via each other radioelectric station, the NbMot words from a Lagrange interpolation of the received polynomial values.
    
    


According to other advantageous aspects, the transmission method comprises one or more of the following features, considered alone or according to all technically possible combinations:

    
    
        each word is divided into a plurality of patterns, and several polynomials are determined using the following equation:
    
    






  
    
      P
      w
    
    ⁡
    
      (
      X
      )
    
  
  =
  
    
      
        (
        
          T
          NbMot
        
        )
      
      w
    
    +
    
      
        ∑
        
          j
          =
          1
        
        
          NbMot
          -
          1
        
      
      ⁢
      
        
          
            (
            
              T
              j
            
            )
          
          w
        
        ×
        
          X
          j
        
      
    
  




with NbMot>1, 0≦w≦v and v≧1 and X designating the variable;

    
    
        NbRepet data packets are successively transmitted during the transmission step;
        the packets are defined as follows:

Paquet(i)=P0((α0)NbRepet-1-i)∥ . . . ∥Pw((αw)NbRepet-1-i)∥ . . . ∥Pv((αv)NbRepet-1-i)

    
    


with NbRepet>1, 0≦i≦NbRepet−1, 0≦w≦v;
(αw)NbRepet-1-i designating the predetermined respective value of the variable to calculate the corresponding polynomial value, and
∥ symbolizing a data concatenation;

    
    
        the method further comprises a step for calculating, via each station receiving said packets, time synchronization information for time synchronizing with the radioelectric station transmitting said packets, from polynomial values received and reception time deviations between the different received packets;
        the NbMot data words have a same size equal to N bits, N being an integer strictly greater than 1;
        one word among the NbMot data words comprises k bit(s) having a predetermined constant value, k being an integer with 1≦k≦N;
        each word comprises at least one pattern, and each pattern represents an element of a mathematical field, such as a Galois field;
        each respective predetermined value, for which a polynomial value is calculated, is a primitive element of said Galois field.
    
    


Another aspect is a network for transmitting information, the network comprising a plurality of radioelectric stations, each station comprising a transmitter and a receiver, the information comprising NbMot data words, NbMot being an integer strictly greater than 1,
wherein each station designed to transmit the information further comprises:

    
    
        first means for determining at least one polynomial of degree NbMot−1, each of the NbMot coefficients of the polynomial corresponding to a respective word, each polynomial having a variable,
        first means for calculating NbRepet polynomial values for each polynomial, NbRepet being an integer strictly greater than 1, each polynomial value being calculated for a respective predetermined value of the variable, and
        means for transmitting the calculated polynomial values to at least one other radioelectric station,
    
    


and in that each station designed to receive the information further comprises:

    
    
        means for receiving polynomial values, and
        second means for determining the NbMot words from a Lagrange interpolation of the received polynomial values.
    
    


According to another advantageous aspect, the transmission network comprises the following feature:

    
    
        each station designed to receive the information further comprises a second means for calculating time synchronization information for time synchronizing with the station designed to transmit the information, from the received polynomial values and reception time deviations between different received data packets, NbRepet data packets being able to be successively transmitted by the station intended to transmit the information.
    
    





BRIEF DESCRIPTION OF THE DRAWINGS
The features and advantages of the described technology will appear upon reading the following description, provided solely as a non-limiting example, and done in reference to the appended drawings, in which:
FIG. 1 is a diagrammatic illustration of a radioelectric station of a transmission network according to an embodiment.
FIG. 2 is a diagrammatic illustration of the information to be transmitted comprising NbMot data words, NbMot being an integer strictly greater than 1.
FIG. 3 is a diagrammatic illustration of the successive transmission of NbRepet data packets, NbRepet being an integer strictly greater than 1.
FIG. 4 is a flowchart of a method for transmitting information according to an embodiment.



DETAILED DESCRIPTION OF CERTAIN INVENTIVE EMBODIMENTS
A transmission network according to an embodiment comprises a plurality of radioelectric stations or radio stations 10, and is for example able to operate by frequency hopping. A radioelectric station 10 is illustrated in FIG. 1.
The transmission network is able to transmit, from one radioelectric station 10 to another radioelectric station 10, information, visible in FIG. 2, comprising NbMot data words T1, T2, . . . , TNbMot, NbMot being an integer strictly greater than 1. The radioelectric station 10 designed to transmit the information is hereinafter called the transmitting station, and the or each radioelectric station 10 designed to receive the information is hereinafter called the receiving station.
The set of frequencies usable by the transmission network comprises a frequency plane, also called an FH plane, for the frequency hopping operation. The FH plane for example belongs to the high frequency (HF) domain, between about 1.5 MHz and about 30 MHz. The FH plane is for example a predetermined frequency plane.
The number of radioelectric stations 10 is for example greater than or equal to three, each station 10 being able to transmit data in conference to the other stations 10 of the network.
As shown in FIG. 1, each station 10 comprises a transmission-reception chain 12, an antenna 14, a frequency synthesizer 16 and a processing unit 18.
The transmission-reception chain 12 is connected to the transmission-reception antenna 14, the frequency synthesizer 16 and the processing unit 18, the frequency synthesizer 16 and the processing unit 18 also being connected to each other.
The transmission-reception chain 12 comprises a device 20 for receiving radioelectric signals from the antenna 14 and a device 22 for transmitting radioelectric signals to the antenna 14. The transmission-reception chain 12 can be controlled in a known manner by the frequency synthesizer 16.
The processing unit 18 comprises an analog-digital converter 24 connected at the output of the reception device 20 of the transmission-reception chain, a microprocessor 26 connected at the output of the analog-digital converter 24, and a memory 28 connected to the microprocessor 26.
The analog-digital converter 24 is able to convert the voltage or current level provided by the reception device 20 into digital signal samples transmitted to the microprocessor 26.
The microprocessor 26 is connected to the frequency synthesizer 16 and the transmission-reception chain 12 for the transmission of radioelectric signals intended for other transmitters-receivers 10 of the transmission network.
In the described embodiment, each station 10 is configured to operate alternatively as a transmitting station and a receiving station.
For the operation of the station 10 as a transmitting station, the memory 28 is able to store a first software 30 for determining at least one polynomial Pw(X), 0≦w≦v, of degree NbMot−1, each of the NbMot coefficients of the polynomial corresponding to a respective word, each polynomial having a variable X. The memory 28 is able to store a first program 32 for calculating NbRepet polynomial values Pw((αw)NbRepet-1-i), 0≦w≦v, 0≦i≦NbRepet−1, for the or each polynomial Pw(X), NbRepet being an integer strictly greater than 1, each polynomial value Pw((αw)NbRepet-1-i) being calculated for a respective predetermined value (αw)NbRepet-1-i of the variable X. The memory 28 is also able to store a transmission program 34, intended for at least one other radioelectric station 10, for transmitting the calculated polynomial values Pw((αw)NbRepet-1-i).
For the operation of the station 10 as a receiving station, the memory 28 is able to store a software 36 for receiving said polynomial values Pw((αw)NbRepet-1-i) transmitted by a transmitting station 10, a second software 38 for determining the NbMot words T1, T2, . . . , TNbMot from a Lagrange interpolation of the received polynomial values Pw((αw)NbRepet-1-m).
The transmission method is for example carried out during a synchronization phase between several radioelectric stations 10. The synchronization phase corresponds to the transmission of a burst of elementary radio packets, i.e., a fixed number of elementary radio packets. The synchronization procedure is used primarily to transmit two types of information between the transmitting station and the receiving station, i.e., a technical datum necessary to decode the information and a time synchronization information t_final, the technical datum for example comprising the address of the transmitter, the address of the recipient, the type of data transmission services, security data, such as an integrity pattern, and cryptographic authentication. The time synchronization information allows the receiving station to position itself in the burst of received packets and to unambiguously deduce the start time of the transmission phase of the information that follows the synchronization phase.
Additionally, the memory 28 is able to store a second software 40 for calculating the time synchronization information t_final with the transmitting station 10, from the received polynomial values Pw((αw)NbRepet-1-m) and reception time deviations between different received data packets Rm, with m an integer such that 0≦m≦NbMot−1, NbRepet data packets Paquet(i), 0≦i≦NbRepet−1, being able to be successively transmitted by the transmitting station 10.
Alternatively, each station 10 is configured to operate only as a transmitting station or only as a receiving station. The memory 28 of the transmitting station 10 then only comprises the first determination software 30, the first calculation software 32 and the transmission software 34. The memory 28 of each receiving station 10 then only comprises the reception software 36 and the second determination software 38. Additionally, the memory 28 of each receiving station 10 comprises the second calculation software 40.
Alternatively, the first determination means 30, the first calculation means 32, the transmission means 34, the reception means 36, the second determination means 38 and the second calculation means 40 are in the form of programmable logic components, or in the form of dedicated integrated circuits.
The method for transmitting the information will now be described in light of FIGS. 2 to 4, FIG. 4 in particular illustrating a flowchart of the transmission method.
During step 100, the transmitting station 10 uses the first determination software 30 to determine at least one polynomial Pw(X), 0≦w≦v, of degree NbMot−1, each of the NbMot coefficients of the polynomial Pw(X) corresponding to a respective word T1, . . . , TNbMot. The or each polynomial Pw(X) comprises a variable X, the variable X preferably being unique for the or each polynomial Pw(X), where w and v are integers.
Each word T1, . . . , TNbMot comprises at least one pattern (Tj)w, 0≦w≦v, j being an integer such that 1≦j≦NbMot, and the or each pattern (Tj)w represents an element of a mathematical field, such as the Galois field.
The NbMot data words T1, T2, . . . , TNbMot for example have the same size equal to N bits, N being an integer strictly greater than 1.
In the described example, one word, such as the first word T1, among the NbMot data words T1, T2, . . . , TNbMot comprises k bit(s) having a predetermined constant value CST, k being an integer with 1≦k≦N, k desirably being strictly greater than 1 and also such that NbRepet−1<2k.
The k bits equal to CST for example represent an element of the Galois field GF(2k). Out of a concern for homogeneity with the rest of the description, k is denoted a(0) such that GF(2k) is equal to GF(2a(0)). α0 is a primitive element of the Galois field)) GF(2a(0)). The N-k other bits of the first word T1 represent the succession of v elements respectively belonging to v bodies with form GF(2a(1)), . . . , GF(2a(v)), v being an integer greater than or equal to 1, such that

    
    
        for any w, 1≦w≦v, a(w)≧k,
    
    






  
    
      
        
          
            ∑
            
              w
              =
              1
            
            v
          
          ⁢
          
            a
            ⁡
            
              (
              w
              )
            
          
        
        =
        
          N
          -
          k
        
      
    
    
      
        (
        1
        )
      
    
  




In the specific case where v is equal to 1, the N-k bits represent an element of the field GF(2N-k).
α1, . . . , αv represent the v respective primitive elements of the v successive Galois bodies GF(2a(1)), . . . , GF(2a(v)).
In the described example embodiment, each word Tj, 1≦j≦NbMot, is divided into v+1 patterns (Tj)w, 0≦w≦v, v being greater than or equal to 1, each pattern (Tj)w being an element of a respective field GF(2a(j)). Each word Tj is then represented as follows:

Tj=(Tj)0∥(Tj)1∥(Tj)2∥ . . . ∥(Tj)v  (2)

In the particular case where the k first bits of the first word T1 are equal to CST, we also have: (T1)0=CST.
The first determination software 30 then determines several polynomials Pw(X), 0≦w≦v and v≧1, using the following equation:




  
    
      
        
          
            P
            w
          
          ⁡
          
            (
            X
            )
          
        
        =
        
          
            
              (
              
                T
                NbMot
              
              )
            
            w
          
          +
          
            
              ∑
              
                j
                =
                1
              
              
                NbMot
                -
                1
              
            
            ⁢
            
              
                
                  (
                  
                    T
                    j
                  
                  )
                
                w
              
              ×
              
                X
                j
              
            
          
        
      
    
    
      
        (
        3
        )
      
    
  




with NbMot>1, 0≦w≦v and v≧1 and X designating the variable.
In the described example embodiment, the coefficients of each polynomial Pw(X) belong to the Galois field GF(2a(w)). Each of the NbMot coefficients (Tj)w of the polynomial Pw(X) then corresponds to a respective pattern.
More specifically, the first determination software 30 then determines the v+1 following polynomials:




  
    
      
        
          
            
              
                
                  
                    
                      
                        
                          
                            P
                            0
                          
                          ⁡
                          
                            (
                            X
                            )
                          
                        
                        =
                        
                          
                            
                              (
                              
                                T
                                NbMot
                              
                              )
                            
                            0
                          
                          +
                          
                            CST
                            ·
                            X
                          
                          +
                          
                            
                              
                                (
                                
                                  T
                                  2
                                
                                )
                              
                              0
                            
                            ·
                            
                              X
                              2
                            
                          
                          +
                          ...
                          +
                        
                      
                       
                    
                    ⁢
                    
                      
                        (
                        
                          T
                          
                            NbMot
                            -
                            1
                          
                        
                        )
                      
                      0
                    
                    ×
                    
                      X
                      
                        NbMot
                        -
                        1
                      
                    
                  
                  ⁢
                  
                    

                  
                  ⁢
                  
                    
                      
                        P
                        1
                      
                      ⁡
                      
                        (
                        X
                        )
                      
                    
                    =
                    
                      
                        
                          (
                          
                            T
                            NbMot
                          
                          )
                        
                        1
                      
                      +
                      
                        
                          
                            (
                            
                              T
                              1
                            
                            )
                          
                          1
                        
                        ·
                        X
                      
                      +
                      
                        
                          
                            (
                            
                              T
                              2
                            
                            )
                          
                          1
                        
                        ·
                        
                          X
                          2
                        
                      
                      +
                      ...
                      +
                    
                  
                
                 
              
              ⁢
              
                
                  (
                  
                    T
                    
                      NbMot
                      -
                      1
                    
                  
                  )
                
                1
              
              ×
              
                X
                
                  NbMot
                  -
                  1
                
              
            
            ⁢
            
              

            
            ⁢
            
              
            
            ⁢
            ...
            ⁢
            
              

            
            ⁢
            
              
                
                  P
                  v
                
                ⁡
                
                  (
                  X
                  )
                
              
              =
              
                
                  
                    (
                    
                      T
                      NbMot
                    
                    )
                  
                  v
                
                +
                
                  
                    
                      (
                      
                        T
                        1
                      
                      )
                    
                    v
                  
                  ·
                  X
                
                +
                
                  
                    
                      (
                      
                        T
                        2
                      
                      )
                    
                    v
                  
                  ·
                  
                    X
                    2
                  
                
                +
                ...
                +
              
            
          
           
        
        ⁢
        
          
            (
            
              T
              
                NbMot
                -
                1
              
            
            )
          
          v
        
        ×
        
          X
          
            NbMot
            -
            1
          
        
      
    
    
      
        (
        4
        )
      
    
  




During step 110, the transmitting station 10 uses the first calculation software 32 to determine NbRepet polynomial values Pw((αw)NbRepet-1-i), 0≦w≦v, 0≦i≦NbRepet−1, for the or each polynomial Pw(X), i being an integer, NbRepet being an integer strictly greater than 1. Each polynomial value Pw((αw)NbRepet-1-i) is calculated for a respective predetermined value of the variable X.
Each respective predetermined value for which a polynomial value Pw((αw)NbRepet-1-i) is calculated is, for example, the primitive element (αw)NbRepet-1-i, 0≦w≦v, 0≦i≦NbRepet−1, of a respective Galois field GF(2a(w)).
The calculation of the polynomial value P0((α0)NbRepet-1-i) corresponds to calculations in the Galois field GF(2a(0)), i.e., in GF(2k). The calculation of the polynomial value P1((α1)NbRepet-1-i) corresponds to calculations in the Galois field GF(2a(1)), and so forth.
During step 120, the transmitting station 10 next uses the transmission software 34 to transmit the calculated polynomial values Pw((αw)NbRepet-1-i) to at least one other receiving station 10. The calculated polynomial values Pw((αw)NbRepet-1-i) are for example transmitted in the form of NbRepet data packets Paquet(i), 0≦i≦NbRepet−1, transmitted successively.
The packets Pacquet(i) are for example defined as follows:

Paquet(i)=P0((α0)NbRepet-1-i)∥ . . . ∥Pw((αw)NbRepet-1-i)∥ . . . ∥Pv((αv)NbRepet-1-i)  (5)

with NbRepet>1, 0≦i≦NbRepet−1, 0≦w≦v,
(αw)NbRepet-1-i designating the predetermined respective value of the variable to calculate the corresponding polynomial value, and
∥ symbolizing a data concatenation.
During step 130, the or each receiving station 10 receives packets Rm containing said polynomial values Pw((αw)NbRepet-1-m). This assumes that each receiving station 10 receives NbMot packets R0, R1, . . . , RNbMot-1, with the convention Rm=Paquet(t(m)), t(m) designating the reception moment of the packet Rm, with 0≦m≦NbMot−1.
The or each receiving station 10 precisely knows the reception time deviations between the different packets R0, R1, . . . , RNbMot-1. These time deviations are denoted d(n), 1≦n≦NbMot−1 and verify the following equations:

t(1)=t(0)+d(1),

t(2)=t(1)+d(2), . . . ,

t(NbMot−1)=t(NbMot−2)+d(NbMot−1)  (6)

In other words, the different reception moments t(m) are defined as follows:




  
    
      
        
          t
          ⁡
          
            (
            m
            )
          
        
        =
        
          
            t
            ⁡
            
              (
              
                NbMot
                -
                1
              
              )
            
          
          -
          
            
              ∑
              
                n
                =
                
                  m
                  +
                  1
                
              
              
                NbMot
                -
                1
              
            
            ⁢
            
              
                d
                ⁡
                
                  (
                  n
                  )
                
              
              ⁢
              
                
              
              ⁢
              or
            
          
        
      
    
    
      
        (
        7
        )
      
    
  
  
    
      
        
          
            
              t
              ⁡
              
                (
                0
                )
              
            
            =
            
              
                t
                ⁡
                
                  (
                  
                    NbMot
                    -
                    1
                  
                  )
                
              
              -
              
                d
                ⁡
                
                  (
                  1
                  )
                
              
              -
              
                d
                ⁡
                
                  (
                  2
                  )
                
              
              -
              ...
              -
              
                d
                ⁡
                
                  (
                  
                    NbMot
                    -
                    1
                  
                  )
                
              
            
          
          ,
          
            

          
          ⁢
          
            
              t
              ⁡
              
                (
                1
                )
              
            
            =
            
              
                t
                ⁡
                
                  (
                  
                    NbMot
                    -
                    1
                  
                  )
                
              
              -
              
                d
                ⁡
                
                  (
                  2
                  )
                
              
              -
              ...
              -
              
                d
                ⁡
                
                  (
                  
                    NbMot
                    -
                    1
                  
                  )
                
              
            
          
          ,
          ...
          ⁢
          
            
          
          ,
          
            

          
          ⁢
          
            
              t
              ⁡
              
                (
                
                  NbMot
                  -
                  2
                
                )
              
            
            =
            
              
                t
                ⁡
                
                  (
                  
                    NbMot
                    -
                    1
                  
                  )
                
              
              -
              
                d
                ⁡
                
                  (
                  
                    NbMot
                    -
                    1
                  
                  )
                
              
            
          
          ,
          and
        
        ⁢
        
          

        
        ⁢
        
          t
          ⁡
          
            (
            
              NbMot
              -
              1
            
            )
          
        
      
    
    
      
        (
        8
        )
      
    
  




Each received packet Rm is then written in the following form:




  
    
      
        
          
            R
            0
          
          =
          
            
              Paquet
              ⁡
              
                (
                
                  t
                  ⁡
                  
                    (
                    0
                    )
                  
                
                )
              
            
            =
            
              
                
                  P
                  0
                
                ⁡
                
                  (
                  
                    
                      (
                      
                        α
                        0
                      
                      )
                    
                    
                      NbRepet
                      -
                      1
                      -
                      
                        t
                        ⁡
                        
                          (
                          0
                          )
                        
                      
                    
                  
                  )
                
              
              ⁢
              
                
              
              ⁢
              
                
                
                  
                
                ⁢
                ...
                ⁢
                
                  
                
                
              
              ⁢
              
                
              
              ⁢
              
                
                  P
                  v
                
                ⁡
                
                  (
                  
                    
                      (
                      
                        α
                        v
                      
                      )
                    
                    
                      NbRepet
                      -
                      1
                      -
                      
                        t
                        ⁡
                        
                          (
                          0
                          )
                        
                      
                    
                  
                  )
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            R
            1
          
          =
          
            
              Paquet
              ⁡
              
                (
                
                  t
                  ⁡
                  
                    (
                    1
                    )
                  
                
                )
              
            
            =
            
              
                
                  P
                  0
                
                ⁡
                
                  (
                  
                    
                      (
                      
                        α
                        0
                      
                      )
                    
                    
                      NbRepet
                      -
                      1
                      -
                      
                        t
                        ⁡
                        
                          (
                          1
                          )
                        
                      
                    
                  
                  )
                
              
              ⁢
              
                
              
              ⁢
              
                
                
                  
                
                ⁢
                ...
                ⁢
                
                  
                
                
              
              ⁢
              
                
              
              ⁢
              
                
                  P
                  v
                
                ⁡
                
                  (
                  
                    
                      (
                      
                        α
                        v
                      
                      )
                    
                    
                      NbRepet
                      -
                      1
                      -
                      
                        t
                        ⁡
                        
                          (
                          1
                          )
                        
                      
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        9
        )
      
    
  
  
    
      
        
          
            
              
                
                  
                    
                      
                        
                      
                      ⁢
                      
                        
                          ...
                          ⁢
                          
                            

                          
                          ⁢
                          
                            
                              R
                              
                                
                                  NbMot
                                  -
                                  1
                                
                                ⁢
                                
                                  
                                
                              
                            
                            =
                            
                              
                                Paquet
                                ⁢
                                
                                  (
                                  
                                    t
                                    ⁡
                                    
                                      (
                                      
                                        NbMot
                                        -
                                        1
                                      
                                      )
                                    
                                  
                                  )
                                
                              
                              =
                              
                                
                                  
                                    P
                                    0
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        (
                                        
                                          α
                                          0
                                        
                                        )
                                      
                                      
                                        NbRepet
                                        -
                                        1
                                        -
                                        
                                          t
                                          ⁡
                                          
                                            (
                                            
                                              NbMot
                                              -
                                              1
                                            
                                            )
                                          
                                        
                                      
                                    
                                    )
                                  
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  
                                  
                                    
                                  
                                  ⁢
                                  ...
                                  ⁢
                                  
                                    
                                  
                                  
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  
                                    P
                                    v
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        (
                                        
                                          α
                                          v
                                        
                                        )
                                      
                                      
                                        NbRepet
                                        -
                                        1
                                        -
                                        
                                          t
                                          ⁡
                                          
                                            (
                                            
                                              NbMot
                                              -
                                              1
                                            
                                            )
                                          
                                        
                                      
                                    
                                    )
                                  
                                
                              
                            
                          
                        
                        ⁢
                        
                          

                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          which
                          ⁢
                          
                            
                          
                          ⁢
                          is
                          ⁢
                          
                            
                          
                          ⁢
                          rewritten
                          ⁢
                          
                            
                          
                          ⁢
                          in
                          ⁢
                          
                            
                          
                          ⁢
                          the
                          ⁢
                          
                            
                          
                          ⁢
                          form
                          ⁢
                          
                            :
                          
                        
                        ⁢
                        
                          

                        
                        ⁢
                        
                          
                            R
                            
                              0
                              ⁢
                              
                                
                              
                            
                          
                          =
                          
                            
                              
                                P
                                0
                              
                              ⁡
                              
                                (
                                
                                  
                                    (
                                    
                                      α
                                      0
                                    
                                    )
                                  
                                  
                                    NbRepet
                                    -
                                    1
                                    -
                                    
                                      t
                                      ⁡
                                      
                                        (
                                        
                                          NbMot
                                          -
                                          1
                                        
                                        )
                                      
                                    
                                    +
                                    
                                      d
                                      ⁡
                                      
                                        (
                                        1
                                        )
                                      
                                    
                                    +
                                    ...
                                    +
                                    
                                      d
                                      ⁡
                                      
                                        (
                                        
                                          NbMot
                                          -
                                          1
                                        
                                        )
                                      
                                    
                                  
                                
                                )
                              
                            
                            ⁢
                            
                              
                              
                                
                              
                              ⁢
                              ...
                              ⁢
                              
                                
                              
                              
                            
                          
                        
                      
                       
                    
                    ⁢
                    
                      
                        P
                        v
                      
                      ⁡
                      
                        (
                        
                          
                            (
                            
                              α
                              v
                            
                            )
                          
                          
                            NbRepet
                            -
                            1
                            -
                            
                              t
                              ⁡
                              
                                (
                                
                                  NbMot
                                  -
                                  1
                                
                                )
                              
                            
                            +
                            
                              d
                              ⁢
                              
                                (
                                1
                                )
                              
                            
                            +
                            ...
                            +
                            
                              d
                              ⁡
                              
                                (
                                
                                  NbMot
                                  -
                                  1
                                
                                )
                              
                            
                          
                        
                        )
                      
                    
                  
                  ⁢
                  
                    

                  
                  ⁢
                  
                    
                      R
                      
                        1
                        ⁢
                        
                          
                        
                      
                    
                    =
                    
                      
                        
                          P
                          0
                        
                        ⁡
                        
                          (
                          
                            
                              (
                              
                                α
                                0
                              
                              )
                            
                            
                              NbRepet
                              -
                              1
                              -
                              
                                t
                                ⁡
                                
                                  (
                                  
                                    NbMot
                                    -
                                    1
                                  
                                  )
                                
                              
                              +
                              
                                d
                                ⁡
                                
                                  (
                                  2
                                  )
                                
                              
                              +
                              ...
                              +
                              
                                d
                                ⁡
                                
                                  (
                                  
                                    NbMot
                                    -
                                    1
                                  
                                  )
                                
                              
                            
                          
                          )
                        
                      
                      ⁢
                      
                        
                        
                          
                        
                        ⁢
                        ...
                        ⁢
                        
                          
                        
                        
                      
                    
                  
                
                 
              
              ⁢
              
                
                  P
                  v
                
                ⁡
                
                  (
                  
                    
                      (
                      
                        α
                        v
                      
                      )
                    
                    
                      NbRepet
                      -
                      1
                      -
                      
                        t
                        ⁡
                        
                          (
                          
                            NbMot
                            -
                            1
                          
                          )
                        
                      
                      +
                      
                        d
                        ⁡
                        
                          (
                          2
                          )
                        
                      
                      +
                      ...
                      +
                      
                        d
                        ⁡
                        
                          (
                          
                            NbMot
                            -
                            1
                          
                          )
                        
                      
                    
                  
                  )
                
              
            
          
        
        
          
            
              
                
                  
                    
                      
                        
                      
                      ⁢
                      
                        ...
                        ⁢
                        
                          

                        
                        ⁢
                        
                          
                            R
                            
                              
                                NbMot
                                -
                                2
                              
                              ⁢
                              
                                
                              
                            
                          
                          =
                          
                            
                              
                                P
                                0
                              
                              ⁡
                              
                                (
                                
                                  
                                    (
                                    
                                      α
                                      0
                                    
                                    )
                                  
                                  
                                    NbRepet
                                    -
                                    1
                                    -
                                    
                                      t
                                      ⁡
                                      
                                        (
                                        
                                          NbMot
                                          -
                                          1
                                        
                                        )
                                      
                                    
                                    +
                                    
                                      d
                                      ⁡
                                      
                                        (
                                        
                                          NbMot
                                          -
                                          1
                                        
                                        )
                                      
                                    
                                  
                                
                                )
                              
                            
                            ⁢
                            
                              
                              
                                
                              
                              ⁢
                              ...
                              ⁢
                              
                                
                              
                              
                            
                          
                        
                      
                       
                    
                    ⁢
                    
                      
                        P
                        v
                      
                      ⁡
                      
                        (
                        
                          
                            (
                            
                              α
                              v
                            
                            )
                          
                          
                            NbRepet
                            -
                            1
                            -
                            
                              t
                              ⁡
                              
                                (
                                
                                  NbMot
                                  -
                                  1
                                
                                )
                              
                            
                            +
                            
                              d
                              ⁡
                              
                                (
                                
                                  NbMot
                                  -
                                  1
                                
                                )
                              
                            
                          
                        
                        )
                      
                    
                  
                  ⁢
                  
                    

                  
                  ⁢
                  
                    
                      R
                      
                        
                          NbMot
                          -
                          1
                        
                        ⁢
                        
                          
                        
                      
                    
                    =
                    
                      
                        
                          P
                          0
                        
                        ⁡
                        
                          (
                          
                            
                              (
                              
                                α
                                0
                              
                              )
                            
                            
                              NbRepet
                              -
                              1
                              -
                              
                                t
                                ⁡
                                
                                  (
                                  
                                    NbMot
                                    -
                                    1
                                  
                                  )
                                
                              
                            
                          
                          )
                        
                      
                      ⁢
                      
                        
                        
                          
                        
                        ⁢
                        ...
                        ⁢
                        
                          
                        
                        
                      
                    
                  
                
                 
              
              ⁢
              
                
                  P
                  v
                
                ⁡
                
                  (
                  
                    
                      (
                      
                        α
                        v
                      
                      )
                    
                    
                      NbRepet
                      -
                      1
                      -
                      
                        t
                        ⁡
                        
                          (
                          
                            NbMot
                            -
                            1
                          
                          )
                        
                      
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        10
        )
      
    
  




The time synchronization information t_final is defined by the equation:

t_final=NbRepet−1−t(NbMot−1)  (11)

The second determination software 38 then determines, by Lagrange interpolation, v+1 polynomials Qw(X) of degree NbMot−1, with 0≦w≦v and v≧1, using the following equation:




  
    
      
        
          
            Q
            w
          
          ⁡
          
            (
            X
            )
          
        
        =
        
          
            
              P
              w
            
            ⁡
            
              (
              
                
                  
                    (
                    
                      α
                      w
                    
                    )
                  
                  t_final
                
                ⁢
                X
              
              )
            
          
          =
          
            
              
                (
                
                  T
                  NbMot
                
                )
              
              w
            
            +
            
              
                ∑
                
                  j
                  =
                  1
                
                
                  NbMot
                  -
                  1
                
              
              ⁢
              
                
                  
                    (
                    
                      α
                      w
                    
                    )
                  
                  
                    t_final
                    ×
                    j
                  
                
                ×
                
                  
                    (
                    
                      T
                      j
                    
                    )
                  
                  w
                
                ×
                
                  X
                  j
                
              
            
          
        
      
    
    
      
        (
        12
        )
      
    
  




According to the preceding, the NbMot packets R0, R1, . . . , RNbMot-1 are then written



  
    
      R
      0
    
    =
    
      
        
          Q
          0
        
        ⁡
        
          (
          
            
              (
              
                α
                0
              
              )
            
            
              
                d
                ⁡
                
                  (
                  1
                  )
                
              
              +
              ...
              +
              
                d
                ⁡
                
                  (
                  
                    NbMot
                    -
                    1
                  
                  )
                
              
            
          
          )
        
      
      ⁢
      
        
      
      ⁢
      
        
        
          
        
        ⁢
        ...
        ⁢
        
          
        
        
      
      ⁢
      
        
      
      ⁢
      
        
          Q
          v
        
        ⁡
        
          (
          
            
              (
              
                α
                v
              
              )
            
            
              
                d
                ⁡
                
                  (
                  1
                  )
                
              
              +
              ...
              +
              
                d
                ⁡
                
                  (
                  
                    NbMot
                    -
                    1
                  
                  )
                
              
            
          
          )
        
      
    
  




  
    
      R
      1
    
    =
    
      
        
          Q
          0
        
        ⁡
        
          (
          
            
              (
              
                α
                0
              
              )
            
            
              
                d
                ⁡
                
                  (
                  2
                  )
                
              
              +
              ...
              +
              
                d
                ⁡
                
                  (
                  
                    NbMot
                    -
                    1
                  
                  )
                
              
            
          
          )
        
      
      ⁢
      
        
      
      ⁢
      
        
        
          
        
        ⁢
        ...
        ⁢
        
          
        
        
      
      ⁢
      
        
      
      ⁢
      
        
          Q
          v
        
        ⁡
        
          (
          
            
              (
              
                α
                v
              
              )
            
            
              
                d
                ⁡
                
                  (
                  2
                  )
                
              
              +
              ...
              +
              
                d
                ⁡
                
                  (
                  
                    NbMot
                    -
                    1
                  
                  )
                
              
            
          
          )
        
      
    
  




  
    
      
    
    ⁢
    ...
  




  
    
      
    
    ⁢
    
      
        R
        
          NbMot
          -
          2
        
      
      =
      
        
          
            Q
            0
          
          ⁡
          
            (
            
              
                (
                
                  α
                  0
                
                )
              
              
                d
                ⁡
                
                  (
                  
                    NbMot
                    -
                    1
                  
                  )
                
              
            
            )
          
        
        ⁢
        
          
        
        ⁢
        
          
          
            
          
          ⁢
          ...
          ⁢
          
            
          
          
        
        ⁢
        
          
        
        ⁢
        
          
            Q
            v
          
          ⁡
          
            (
            
              
                (
                
                  α
                  v
                
                )
              
              
                d
                ⁡
                
                  (
                  
                    NbMot
                    -
                    1
                  
                  )
                
              
            
            )
          
        
      
    
  




  
    
      
    
    ⁢
    
      
        R
        
          NbMot
          -
          1
        
      
      =
      
        
          
            Q
            0
          
          ⁡
          
            (
            1
            )
          
        
        ⁢
        
          
        
        ⁢
        
          
          
            
          
          ⁢
          ...
          ⁢
          
            
          
          
        
        ⁢
        
          
        
        ⁢
        
          
            Q
            v
          
          ⁡
          
            (
            1
            )
          
        
      
    
  



In the described example embodiment, the coefficients of each polynomial Qw(X) belong to the Galois field GF(2a(w)).
More specifically, the second determination software 38 then determines the v+1 following polynomials:




  
    
      
        
          
            Q
            0
          
          ⁡
          
            (
            X
            )
          
        
        =
        
          
            
              (
              
                T
                NbMot
              
              )
            
            0
          
          +
          
            
              
                CST
                ⁡
                
                  (
                  
                    α
                    0
                  
                  )
                
              
              t_final
            
            ⁢
            X
          
          +
          
            
              
                (
                
                  T
                  2
                
                )
              
              0
            
            ⁢
            
              
            
            ⁢
            
              
                (
                
                  α
                  0
                
                )
              
              
                t_final
                ×
                2
              
            
            ⁢
            
              
            
            ⁢
            
               
              
                
                  
                    X
                    2
                  
                  +
                  ...
                  +
                  
                    
                      
                        (
                        
                          T
                          
                            NbMot
                            -
                            1
                          
                        
                        )
                      
                      0
                    
                    ⁢
                    
                      
                        (
                        
                          α
                          0
                        
                        )
                      
                      
                        t_final
                        ×
                        
                          (
                          
                            NbMot
                            -
                            1
                          
                          )
                        
                      
                    
                    ⁢
                    
                      X
                      
                        NbMot
                        -
                        1
                      
                    
                    ⁢
                    
                      

                    
                    ⁢
                    
                      
                        Q
                        1
                      
                      ⁡
                      
                        (
                        X
                        )
                      
                    
                  
                
                =
                
                  
                    
                      (
                      
                        T
                        NbMot
                      
                      )
                    
                    1
                  
                  +
                  
                    
                      
                        (
                        
                          T
                          1
                        
                        )
                      
                      1
                    
                    ⁢
                    
                      
                        (
                        
                          α
                          1
                        
                        )
                      
                      t_final
                    
                    ⁢
                    X
                  
                  +
                  
                    
                      
                        (
                        
                          T
                          2
                        
                        )
                      
                      1
                    
                    ⁢
                    
                      
                        (
                        
                          α
                          1
                        
                        )
                      
                      
                        t_final
                        ×
                        2
                      
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                       
                      
                        
                          
                            X
                            2
                          
                          +
                          ...
                          +
                          
                            
                              
                                (
                                
                                  T
                                  
                                    NbMot
                                    -
                                    1
                                  
                                
                                )
                              
                              1
                            
                            ⁢
                            
                              
                                (
                                
                                  α
                                  1
                                
                                )
                              
                              
                                t_final
                                ×
                                
                                  (
                                  
                                    NbMot
                                    -
                                    1
                                  
                                  )
                                
                              
                            
                            ⁢
                            
                              X
                              
                                NbMot
                                -
                                1
                              
                            
                            ⁢
                            
                              

                            
                            ⁢
                            
                              
                            
                            ⁢
                            ...
                            ⁢
                            
                              

                            
                            ⁢
                            
                              
                                Q
                                v
                              
                              (
                              
                                
                              
                              ⁢
                              X
                              )
                            
                          
                        
                        =
                        
                          
                        
                        ⁢
                        
                          
                            
                              (
                              
                                T
                                NbMot
                              
                              )
                            
                            v
                          
                          +
                          
                            
                          
                          ⁢
                          
                            
                              
                                (
                                
                                  T
                                  1
                                
                                )
                              
                              v
                            
                            ⁢
                            
                              
                            
                            ⁢
                            
                              
                                (
                                
                                  α
                                  v
                                
                                )
                              
                              t_final
                            
                            ⁢
                            
                              
                            
                            ⁢
                            X
                          
                          +
                          
                            
                          
                          ⁢
                          
                            
                              
                                (
                                
                                  T
                                  2
                                
                                )
                              
                              v
                            
                            ⁢
                            
                              
                            
                            ⁢
                            
                              
                                (
                                
                                  α
                                  v
                                
                                )
                              
                              
                                t_final
                                ×
                                2
                              
                            
                            ⁢
                            
                              
                            
                            ⁢
                            
                               
                              
                                
                              
                              ⁢
                              
                                
                                  X
                                  2
                                
                                +
                                ...
                                +
                              
                               
                            
                            ⁢
                            
                               
                              
                                
                                  
                                    (
                                    
                                      T
                                      
                                        NbMot
                                        -
                                        1
                                      
                                    
                                    )
                                  
                                  v
                                
                                ⁢
                                
                                  
                                    (
                                    
                                      α
                                      v
                                    
                                    )
                                  
                                  
                                    t_final
                                    ×
                                    
                                      (
                                      
                                        NbMot
                                        -
                                        1
                                      
                                      )
                                    
                                  
                                
                                ⁢
                                
                                  X
                                  
                                    NbMot
                                    -
                                    1
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        13
        )
      
    
  




Each coefficient of the polynomial Qw(X) for the term Xj of degree j is denoted Coeff(Qw(X), j), and each polynomial Qw(X) is then written:




  
    
      
        
          
            Q
            w
          
          ⁡
          
            (
            X
            )
          
        
        =
        
          
            
              (
              
                T
                NbMot
              
              )
            
            w
          
          +
          
            
              ∑
              
                j
                =
                1
              
              
                NbMot
                -
                1
              
            
            ⁢
            
              
                Coeff
                ⁡
                
                  (
                  
                    
                      
                        Q
                        w
                      
                      ⁡
                      
                        (
                        X
                        )
                      
                    
                    ,
                    j
                  
                  )
                
              
              ⁢
              
                X
                j
              
            
          
        
      
    
    
      
        (
        14
        )
      
    
  




The second calculation software 40 then calculates the time synchronization information t_final using the coefficient Coeff(Q0(X), 1), the latter verifying, according to equations (12) and (14), the following equation:

Coeff(Q0(X),1)=(α0)t_final(T1)0=(α0)t_finalCST  (15)

Consequently, the quantity (α0)t_final is found by calculating the quantity Coeff(Q0(X),1)/CST in the Galois field GF(2k). The value k being relatively small, the second calculation software 40 is then able to find the time synchronization information t_final from the value (α0)t_final, for example using a match table between the elements (α0)m and m, or via an exhaustive search on the few possible values of the time synchronization information t_final.
The receiving station 10 then knows, at the end of the transmission of the NbRepet−1 data packets Paquet(m) for example corresponding to a synchronization phase S, to position itself exactly on a successive communication phase F, as shown in FIG. 3. The communication phase F for example allows the transmission of a useful information flow at the end of the synchronization phase S.
During step 150, the second determination software 38 lastly determines the value of each of the NbMot words T1, T2, . . . , TNbMot from the Lagrange interpolation of the received polynomial values Pw((αw)NbRepet-1-i).
From the coefficients of each of the v+1 polynomials Qw(X), 0≦w≦v, the second determination software 38 obtains the word TNbMot equivalent to:

TNbMot=(TNbMot)0∥(TNbMot)1∥(TNbMot)2∥ . . . ∥(TNbMot)v  (16)

For the other words Tj, 0≦j≦NbMot−1, the second determination software 38, knowing the value (α0)t_final, determines each pattern (Tj)w from each polynomial Qw(X) using the following equation:




  
    
      
        
          
            (
            
              T
              j
            
            )
          
          w
        
        =
        
          
            Coeff
            ⁡
            
              (
              
                
                  
                    Q
                    w
                  
                  ⁡
                  
                    (
                    X
                    )
                  
                
                ,
                j
              
              )
            
          
          
            
              (
              
                α
                w
              
              )
            
            
              t_final
              ×
              j
            
          
        
      
    
    
      
        (
        17
        )
      
    
  




the value of (αw)t_final being calculated from the known value of the primitive element αw and the value of the time synchronization information t_final previously calculated.
In other words, each word Tj is, according to equations (2) and (17), determined as follows:




  
    
      
        
          T
          j
        
        =
        
          
            
              Coeff
              ⁡
              
                (
                
                  
                    
                      Q
                      0
                    
                    ⁡
                    
                      (
                      X
                      )
                    
                  
                  ,
                  j
                
                )
              
            
            
              
                (
                
                  α
                  0
                
                )
              
              
                t_final
                ×
                j
              
            
          
          ⁢
          
            
            
              
                
                  Coeff
                  ⁡
                  
                    (
                    
                      
                        
                          Q
                          1
                        
                        ⁡
                        
                          (
                          X
                          )
                        
                      
                      ,
                      j
                    
                    )
                  
                
                
                  
                    (
                    
                      α
                      1
                    
                    )
                  
                  
                    t_final
                    ×
                    j
                  
                
              
              ⁢
              
                
                
                  ...
                  ⁢
                  
                    
                    
                      
                        Coeff
                        ⁡
                        
                          (
                          
                            
                              
                                Q
                                v
                              
                              ⁡
                              
                                (
                                X
                                )
                              
                            
                            ,
                            j
                          
                          )
                        
                      
                      
                        
                          (
                          
                            α
                            v
                          
                          )
                        
                        
                          t_final
                          ×
                          j
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        18
        )
      
    
  




As an example, the transmission method is described below in the particular case where NbMot is equal to 2 and the integer number v is equal to 1.
The information that must be transmitted has a size of 2×N bits. Assuming that the constant value CST is located on the first k bits of the first word T1, the first and second words to be transmitted T1, T2 are then written:

T1=CST∥(T1)1  (19)

where CST is an element of the Galois field GF(2k) and (T1)1 is an element of the Galois field GF(2N-k), and

T2=(T2)0∥(T2)1  (20)

where (T2)0 is an element of the Galois field GF(2k) and (T2)1 is an element of the Galois field GF(2N-k).
The first determination software 30 then determines the following two polynomials P0(X) and P1(X) of degree 1:

P0(X)=(T2)0+CST×X 

P1(X)=(T2)1+(T1)1×X 

P0(X)=(T2)0+CST·X 

P1(X)=(T2)1+(T1)1·X  (21)

P0(X) being defined in the Galois field GF(2k) and P1(X) being defined in the Galois field GF(2N-k).
The transmitted NbRepet packets Paquet(i) are then the following:




  
    
      
        
          Paquet
          ⁡
          
            (
            0
            )
          
        
        =
        
          
            
              P
              0
            
            ⁡
            
              (
              
                
                  (
                  
                    α
                    0
                  
                  )
                
                
                  NbRepet
                  -
                  1
                
              
              )
            
          
          ⁢
          
            
            
              
                
                  
                    P
                    1
                  
                  ⁡
                  
                    (
                    
                      
                        (
                        
                          α
                          1
                        
                        )
                      
                      
                        NbRepet
                        -
                        1
                      
                    
                    )
                  
                
                ⁢
                
                  

                
                ⁢
                
                  Paquet
                  ⁡
                  
                    (
                    1
                    )
                  
                
              
              =
              
                
                  
                    P
                    0
                  
                  ⁡
                  
                    (
                    
                      
                        (
                        
                          α
                          0
                        
                        )
                      
                      
                        NbRepet
                        -
                        2
                      
                    
                    )
                  
                
                ⁢
                
                  
                  
                    
                      
                        P
                        1
                      
                      ⁡
                      
                        (
                        
                          
                            (
                            
                              α
                              1
                            
                            )
                          
                          
                            NbRepet
                            -
                            2
                          
                        
                        )
                      
                    
                    ⁢
                    
                      

                    
                    ⁢
                    ...
                    ⁢
                    
                      

                    
                    ⁢
                    
                      
                        Paquet
                        ⁡
                        
                          (
                          
                            NbRepet
                            -
                            2
                          
                          )
                        
                      
                      =
                      
                        
                          
                            P
                            0
                          
                          ⁡
                          
                            (
                            
                              α
                              0
                            
                            )
                          
                        
                        ⁢
                        
                          
                          
                            
                              
                                
                                  P
                                  1
                                
                                ⁡
                                
                                  (
                                  
                                    α
                                    1
                                  
                                  )
                                
                              
                              ⁢
                              
                                

                              
                              ⁢
                              
                                Paquet
                                ⁡
                                
                                  (
                                  
                                    NbRepet
                                    -
                                    1
                                  
                                  )
                                
                              
                            
                            =
                            
                              
                                
                                  P
                                  0
                                
                                ⁡
                                
                                  (
                                  1
                                  )
                                
                              
                              ⁢
                              
                                
                                
                                  
                                    P
                                    1
                                  
                                  ⁡
                                  
                                    (
                                    1
                                    )
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        22
        )
      
    
  




In reception, it is then assumed that the receiving station 10 receives NbMot packets, i.e., 2 packets R0 and R1, at respective moments t0 and t1, with t1 equal to t0+d, where d represents the time deviation between the two received packets, that time deviation d being known by the receiving station 10.
The first received packet R0 verifies:

R0=Paquet(t0)=P0((α0)NbRepet-1-t0)∥P1((α1)NbRepet-1-t0)   (23)

which is written in the form

R0=P0((α0)NbRepet-1-t1(α0)d)∥P1((α1)NbRepet-1-t1(α1)d)=Q0((α0)d)∥Q1((α1)d)   (24)

given that

t1=t0+d   (25)

The second received packet R1 verifies:

R1=Paquet(t1)=P0((α0)NbRepet-1-t1)∥P1((α1)NbRepet-1-t1)   (26)

which is written in the form

R1=P0((α0)NbRepet-1-t1(α0)0)∥P1((α1)NbRepet-1-t1(α1)0)=Q0(1)∥Q1(1)   (27)

The receiving station 10 reconstructs the polynomials by Lagrange interpolation:

Q0(X)=Coeff(Q0(X),0)+Coeff(Q0(X),1)×X=P0((α0)NbRepet-1-t1×X)  (28)

Q1(X)=Coeff(Q1(X),0)+Coeff(Q1(X),1)×X=P1((α1)NbRepet-1-t1×X)  (29)

The receiving station 10 deduces, by polynomial identification, that

T2=Coeff(Q0(X),0)∥Coeff(Q1(X),0)   (30)

and that

Coeff(Q0(X),1)=(α0)NbRepet-1-t1(T1)0=(α0)NbRepet-1-t1CST   (31)

then that

Coeff(Q1(X),1)=(α1)NbRepet-1-t1(T1)1   (32)

The receiving station 10 then deduces the value of NbRepet−1−t1 from equation (31), then the value of the pattern (T1)1 from the following equation (32):




  
    
      
        
          
            (
            
              T
              1
            
            )
          
          1
        
        =
        
          
            Coeff
            ⁡
            
              (
              
                
                  
                    Q
                    1
                  
                  ⁡
                  
                    (
                    X
                    )
                  
                
                ,
                1
              
              )
            
          
          
            
              (
              
                α
                1
              
              )
            
            
              NbRepet
              -
              1
              -
              
                t
                ⁢
                
                  
                
                ⁢
                1
              
            
          
        
      
    
    
      
        (
        33
        )
      
    
  




The transmission method and the transmission network according to the described technology makes it possible to transmit the information comprising the data words T1, . . . , TnbMot, between a transmitting station 10 and one or more receiving stations 10 via the transmission of only NbRepet data packets Paquet(i), while guaranteeing the desired quality of service.
As a comparison, for a same quality of service and for a same size of the transmitted packets, the transmission method according to the described technology requires a number of transmitted packets approximately 2.5 times lower than the number of transmitted packets necessary with the standard transmission method.
Furthermore, the transmission method according to the described technology also makes it possible to calculate the time synchronization information t_final quite simply.
One can thus see that the transmission method and network according to the described technology makes it possible to reduce the quantity of data transmitted by the transmitting station to the receiving station(s), to convey said information comprising the data words T1, . . . , TnbMot, while offering the same quality of service as the transmission method of the state of the art.
The transmission network according to the invention is thus particularly faster for transmitting the information.
While there have been shown and described and pointed out the fundamental novel features of the invention as applied to certain inventive embodiments, it will be understood that the foregoing is considered as illustrative only of the principles of the invention and not intended to be exhaustive or to limit the invention to the precise forms disclosed. Naturally, modifications and variations are within the scope of the invention as determined by the appended claims when interpreted in accordance with the breadth to which they are entitled.

## Claims

1. A method for transmitting information between a plurality of radioelectric stations, each of the radioelectric stations comprising a transmitter and a receiver, the information comprising NbMot words of data, NbMot being an integer strictly greater than 1, the method comprising:
determining at least one polynomial of degree NbMot−1, each of the NbMot coefficients of the polynomial corresponding to a respective one of the words, the polynomial having a variable;
calculating NbRepet polynomial values for the polynomial, NbRepet being an integer strictly greater than 1, each of the polynomial values being calculated for a respective predetermined value of the variable;
transmitting the calculated polynomial values from a first radioelectric station to at least one second radioelectric station, wherein NbRepet data packets are successively transmitted during the transmission;
receiving polynomial values by the second radioelectric station;
determining, via the second radioelectric station, the words from a Lagrange interpolation of the received polynomial values; and
calculating, via the second radioelectric station receiving the packets, time synchronization information for time synchronizing with the first radioelectric station transmitting the packets, from the received polynomial values and reception time deviations between the different received packets.

2. The method according to claim 1, wherein each of the words is divided into a plurality of patterns, and wherein several of the polynomials are determined using the following equation:




  
    
      P
      w
    
    ⁡
    
      (
      X
      )
    
  
  =
  
    
      
        (
        
          T
          NbMot
        
        )
      
      w
    
    +
    
      
        ∑
        
          j
          =
          1
        
        
          NbMot
          -
          1
        
      
      ⁢
      
        
          
            (
            
              T
              j
            
            )
          
          w
        
        ×
        
          X
          j
        
      
    
  



with NbMot>1, 0≦w≦v and v≧1, j being an index designating a respective one of the words and X designating the variable.

3. The method according to claim 1, wherein the packets are defined as follows:


Paquet(i)=P0((α0)NbRepet-1-i)∥ . . . ∥Pw((αw)NbRepet-1-i)∥ . . . ∥Pv((αv)NbRepet-1-i)

with NbRepet>1, 0≦i≦NbRepet−1, 0≦w≦v, Pw designating a respective polynomial, (αw)NbRepet−1-i designating the predetermined respective value of the variable to calculate the corresponding polynomial value, and ∥ symbolizing a data concatenation.

4. The method according to claim 1, wherein the data words each have the same size equal to N bits, N being an integer strictly greater than 1.

5. The method according to claim 4, wherein one word of the data words comprises k bit(s) having a predetermined constant value, k being an integer with 1≦k≦N.

6. The method according to claim 1, wherein each of the words comprises at least one pattern, and wherein the pattern represents an element of a mathematical field.

7. The method according to claim 6, wherein each of the respective predetermined values for which a polynomial value is calculated is a primitive element of the Galois field.

8. A network for transmitting information, the network comprising a plurality of radioelectric stations, each of the radioelectric stations comprising a transmitter and a receiver, the information comprising NbMot words of data, NbMot being an integer strictly greater than 1, wherein each of the radioelectric stations designed to transmit the information further comprises:
means for determining at least one polynomial of degree NbMot−1, each of the NbMot coefficients of the polynomial corresponding to a respective one of the words, the polynomial having a variable;
means for calculating NbRepet polynomial values for the polynomial, NbRepet being an integer strictly greater than 1, each of the polynomial values being calculated for a respective predetermined value of the variable; and
means for transmitting the calculated polynomial values to at least one other radioelectric station,
wherein each of the stations designed to receive the information further comprises:
means for receiving the polynomial values;
means for determining the words from a Lagrange interpolation of the received polynomial values;
means for calculating time synchronization information for time synchronizing with the station designed to transmit the information, from the received polynomial values; and
reception time deviations between different received packets, the station designed to transmit the information configured to successively transmit NbRepet data packets.