Abstract:
An approach to dividing syndrome calculations into two steps and serially processing them requires a long time for the syndrome calculations with respect to an entire decoding process. Therefore, there is disclosed an error correction decoding circuit for a playing signal having a code sequence having a decoding unit generating first decoded signal and second decoded signal based on the code sequence and an error correction unit performing error correction for the second signal in response to the first signal.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to a decoding method and a decoding circuit, and more particularly to a decoding method and a decoding circuit which decode position information (addresses) on an optical disk. 
     2. Description of the Related Art 
     In recent years, a recording and playing apparatus using an optical disk has been paid attention to because of reasons of its compactness, high capacity and the like. 
       FIG. 5  shows an overall view of such a conventional recording and playing apparatus. An optical disk  100  is rotatably driven by a spindle motor  101  at the time of recording and playing operations, and data is recorded in the disk  100  or data is played from the disk  100  by an optical pickup. 
     A pickup  102  has a laser diode which becomes a laser source, a photodetector for detecting a reflected light, and an objective lens which becomes an output end of a laser light, and illuminates the laser light onto a disk recording surface. This pickup  102  can move in a radial direction of the disk by a thread mechanism  103 . Reflected light information from the disk  100  is detected by the photodetector, converted into an electrical signal depending on an amount of received light, and supplied to a matrix circuit  105 . 
     The matrix circuit  105  generates a play data signal, a focus error signal for servo control, a tracking error signal, wobble information and the like. The play data signal outputted from the matrix circuit  105  is supplied to a playing/recording control circuit  106 , the focus error signal and the tracking error signal are supplied to a servo control circuit  107 , and the wobble information is supplied to a wobble decoding circuit  108 , respectively. (The wobble information will be described later.) 
     If the disk  100  is a rewritable disk, the wobble decoding circuit  108  supplies ADIP information which has been MSK-demodulated or HMW-demodulated by using the wobble information, to an address decoding circuit  109 . The address decoding circuit  109  performs an error detection/correction process with respect to the ADIP information and decodes addresses on the disk. 
     At the time of recording, recorded data is transferred from an AV system  114 , sent to and buffered in a memory of an ECC/scrambling circuit  110 , and addition of an error correction code, a scrambling process, addition of a subcode or the like is performed thereto. Moreover, data which has been ECC-encoded and subjected to the scrambling process is applied with modulation of an RLL (1-7) PP scheme in a modulation and demodulation circuit  111  and supplied as an encoded data to the playing/recording control circuit  106 . 
     The playing/recording control circuit  106  sends a laser pulse converted from the encoded data based on the addresses on the disk, to a laser driver  104 . The laser driver  104  performs laser emission driving and forms pits depending on the recorded data on the disk  100 . 
     A spindle control circuit  112  controls to rotate the spindle motor  101 . The spindle control circuit  112  obtains a clock generated in a PLL process with respect to a wobble signal as current rotation speed information on the spindle motor  101 , compares this information with predetermined speed information, and thereby generates a spindle error signal. 
     The above described recording and playing operations are controlled by a system controller  113  formed with a microcomputer. The system controller  113  executes recording and playing processes based on commands from the AV system  114 . 
     Here, on a recording film of the optical disk, a groove for guiding the laser light is formed along a recording track. This groove winds its way (wobbling) with a substantially constant period, which is referred to as “wobble”. The wobble is locally applied with phase modulation, and position information (addresses) on the optical disk, which is required at the time of recording and playing, is embedded therein. These addresses are coded based on a predetermined generation polynomial, converted into wobble patterns having different periods and amplitudes, and embedded on the optical disk. 
     In order to decode a code sequence of the addresses embedded on the optical disk in this way and extract the address information, it is necessary to perform error detection and correction with respect to the code sequence by first syndrome calculations, and perform second syndrome calculations to extract the address information from the code sequence in which the error detection and correction have been performed. 
     A flow of decoding the above described code sequence into the address information will be described in detail. A code sequence C 0  to C(i−1) is obtained from a bit column of ADIP (Address in Pre groove) obtained from the wobble patterns on the optical disk. Here, it is assumed that the code sequence C 0  to C(i−1) is composed in units of nibbles (4 bits) per symbol. If it is assumed that there is no error in this code sequence C 0  to C(i−1), C 0  to C(i−1) are used to extract address information n 0  to n(j−2). n 0  to n(j−2) can be calculated by applying C 0  to C(i−1) to a second syndrome calculation formula (1) for extracting the address information. 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Formula 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       1 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             S 
                             u 
                           
                           = 
                           
                             
                               ∑ 
                               
                                 t 
                                 = 
                                 0 
                               
                               
                                 i 
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               
                                 C 
                                 t 
                               
                               * 
                               
                                 
                                   ( 
                                   
                                     α 
                                     u 
                                   
                                   ) 
                                 
                                 
                                   
                                     ( 
                                     
                                       i 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                   - 
                                   t 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             ( 
                             
                               i 
                               - 
                               j 
                             
                             ) 
                           
                           ⁢ 
                           
                             ≤ 
                             _ 
                           
                           ⁢ 
                           u 
                           ⁢ 
                           
                             ≤ 
                             _ 
                           
                           ⁢ 
                           
                             ( 
                             
                               i 
                               - 
                               2 
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                           
                             n 
                             
                               u 
                               - 
                               
                                 ( 
                                 
                                   i 
                                   - 
                                   j 
                                 
                                 ) 
                               
                             
                           
                           = 
                           
                             S 
                             u 
                           
                         
                       
                       
                         
                           
                             ( 
                             
                               i 
                               - 
                               j 
                             
                             ) 
                           
                           ⁢ 
                           
                             ≤ 
                             _ 
                           
                           ⁢ 
                           u 
                           ⁢ 
                           
                             ≤ 
                             _ 
                           
                           ⁢ 
                           
                             ( 
                             
                               i 
                               - 
                               2 
                             
                             ) 
                           
                         
                       
                     
                   
                   } 
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     In the formula (1), i denotes a code length, j denotes an information length, and u denotes a syndrome number. Moreover, α denotes a primitive element on a Galois field (2 4 ). 
     If the code length i=15 and the information length j=9 are assumed, the above described formula (1) is developed into a formula (2). As shown in  FIG. 1 , decoding into address information n 0  to n 8  is performed by assigning a code sequence C 0  to C 14  to a formula (4). 
     
       
         
           
             
               
                 
                   [ 
                   
                     Formula 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             n 
                             0 
                           
                           = 
                           
                             
                               S 
                               6 
                             
                             = 
                             
                               
                                 
                                   
                                     C 
                                     0 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       6 
                                     
                                     ) 
                                   
                                 
                                 14 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     1 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       6 
                                     
                                     ) 
                                   
                                 
                                 13 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       6 
                                     
                                     ) 
                                   
                                 
                                 12 
                               
                               + 
                               … 
                               + 
                               
                                 
                                   
                                     C 
                                     12 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       6 
                                     
                                     ) 
                                   
                                 
                                 2 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     13 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       6 
                                     
                                     ) 
                                   
                                 
                                 1 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     14 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       6 
                                     
                                     ) 
                                   
                                 
                                 0 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             n 
                             1 
                           
                           = 
                           
                             
                               S 
                               7 
                             
                             = 
                             
                               
                                 
                                   
                                     C 
                                     0 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       7 
                                     
                                     ) 
                                   
                                 
                                 14 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     1 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       7 
                                     
                                     ) 
                                   
                                 
                                 13 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       7 
                                     
                                     ) 
                                   
                                 
                                 12 
                               
                               + 
                               … 
                               + 
                               
                                 
                                   
                                     C 
                                     12 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       7 
                                     
                                     ) 
                                   
                                 
                                 2 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     13 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       7 
                                     
                                     ) 
                                   
                                 
                                 1 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     14 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       7 
                                     
                                     ) 
                                   
                                 
                                 0 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             n 
                             2 
                           
                           = 
                           
                             
                               S 
                               8 
                             
                             = 
                             
                               
                                 
                                   
                                     C 
                                     0 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       8 
                                     
                                     ) 
                                   
                                 
                                 14 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     1 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       8 
                                     
                                     ) 
                                   
                                 
                                 13 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       8 
                                     
                                     ) 
                                   
                                 
                                 12 
                               
                               + 
                               … 
                               + 
                               
                                 
                                   
                                     C 
                                     12 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       8 
                                     
                                     ) 
                                   
                                 
                                 2 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     13 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       8 
                                     
                                     ) 
                                   
                                 
                                 1 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     14 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       8 
                                     
                                     ) 
                                   
                                 
                                 0 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             n 
                             3 
                           
                           = 
                           
                             
                               S 
                               9 
                             
                             = 
                             
                               
                                 
                                   
                                     C 
                                     0 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       9 
                                     
                                     ) 
                                   
                                 
                                 14 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     1 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       9 
                                     
                                     ) 
                                   
                                 
                                 13 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       9 
                                     
                                     ) 
                                   
                                 
                                 12 
                               
                               + 
                               … 
                               + 
                               
                                 
                                   
                                     C 
                                     12 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       9 
                                     
                                     ) 
                                   
                                 
                                 2 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     13 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       9 
                                     
                                     ) 
                                   
                                 
                                 1 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     14 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       9 
                                     
                                     ) 
                                   
                                 
                                 0 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             n 
                             4 
                           
                           = 
                           
                             
                               S 
                               10 
                             
                             = 
                             
                               
                                 
                                   
                                     C 
                                     0 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       10 
                                     
                                     ) 
                                   
                                 
                                 14 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     1 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       10 
                                     
                                     ) 
                                   
                                 
                                 13 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       10 
                                     
                                     ) 
                                   
                                 
                                 12 
                               
                               + 
                               … 
                               + 
                               
                                 
                                   
                                     C 
                                     12 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       10 
                                     
                                     ) 
                                   
                                 
                                 2 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     13 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       10 
                                     
                                     ) 
                                   
                                 
                                 1 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     14 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       10 
                                     
                                     ) 
                                   
                                 
                                 0 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             n 
                             5 
                           
                           = 
                           
                             
                               S 
                               11 
                             
                             = 
                             
                               
                                 
                                   
                                     C 
                                     0 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       11 
                                     
                                     ) 
                                   
                                 
                                 14 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     1 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       11 
                                     
                                     ) 
                                   
                                 
                                 13 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       11 
                                     
                                     ) 
                                   
                                 
                                 12 
                               
                               + 
                               … 
                               + 
                               
                                 
                                   
                                     C 
                                     12 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       11 
                                     
                                     ) 
                                   
                                 
                                 2 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     13 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       11 
                                     
                                     ) 
                                   
                                 
                                 1 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     14 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       11 
                                     
                                     ) 
                                   
                                 
                                 0 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             n 
                             6 
                           
                           = 
                           
                             
                               S 
                               12 
                             
                             = 
                             
                               
                                 
                                   
                                     C 
                                     0 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       12 
                                     
                                     ) 
                                   
                                 
                                 14 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     1 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       12 
                                     
                                     ) 
                                   
                                 
                                 13 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       12 
                                     
                                     ) 
                                   
                                 
                                 12 
                               
                               + 
                               … 
                               + 
                               
                                 
                                   
                                     C 
                                     12 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       12 
                                     
                                     ) 
                                   
                                 
                                 2 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     13 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       12 
                                     
                                     ) 
                                   
                                 
                                 1 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     14 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       12 
                                     
                                     ) 
                                   
                                 
                                 0 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             n 
                             7 
                           
                           = 
                           
                             
                               S 
                               13 
                             
                             = 
                             
                               
                                 
                                   
                                     C 
                                     0 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       13 
                                     
                                     ) 
                                   
                                 
                                 14 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     1 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       13 
                                     
                                     ) 
                                   
                                 
                                 13 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       13 
                                     
                                     ) 
                                   
                                 
                                 12 
                               
                               + 
                               … 
                               + 
                               
                                 
                                   
                                     C 
                                     12 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       13 
                                     
                                     ) 
                                   
                                 
                                 2 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     13 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       13 
                                     
                                     ) 
                                   
                                 
                                 1 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     14 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       13 
                                     
                                     ) 
                                   
                                 
                                 0 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             n 
                             8 
                           
                           = 
                           
                             C 
                             0 
                           
                         
                       
                     
                   
                   } 
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     On the other hand, if it is assumed that there is an error in the above described code sequence C 0  to C(i−1), a first syndrome calculation (3) is performed to perform the error detection and correction with respect to C 0  to C(i−1), and subsequently, a second syndrome calculation (4) is performed to extract the address information n 0  to n(j−2) from a code sequence C′ 0  to C′(i−1) applied with the error correction. 
     First, C 0  to C(i−1) are applied to the first syndrome calculation (3) and thereby syndrome values S 0  to S 5  for calculating error position and error number information required for the error detection and correction are obtained. The error detection and correction are performed by using these syndrome values. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Formula 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     3 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         
                           S 
                           v 
                         
                         = 
                         
                           
                             ∑ 
                             
                               t 
                               = 
                               0 
                             
                             
                               i 
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             
                               C 
                               t 
                             
                             * 
                             
                               
                                 ( 
                                 
                                   α 
                                   v 
                                 
                                 ) 
                               
                               
                                 
                                   ( 
                                   
                                     i 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                                 - 
                                 t 
                               
                             
                           
                         
                       
                     
                     
                       
                         0 
                         ⁢ 
                         
                           ≤ 
                           _ 
                         
                         ⁢ 
                         v 
                         ⁢ 
                         
                           ≤ 
                           _ 
                         
                         ⁢ 
                         
                           ( 
                           
                             i 
                             - 
                             j 
                             - 
                             1 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
             
               
                 
                   [ 
                   
                     Formula 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     4 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             S 
                             u 
                             ′ 
                           
                           = 
                           
                             
                               ∑ 
                               
                                 t 
                                 = 
                                 0 
                               
                               
                                 i 
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               
                                 C 
                                 t 
                                 ′ 
                               
                               * 
                               
                                 
                                   ( 
                                   
                                     α 
                                     u 
                                   
                                   ) 
                                 
                                 
                                   
                                     ( 
                                     
                                       i 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                   - 
                                   t 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             ( 
                             
                               i 
                               - 
                               j 
                             
                             ) 
                           
                           ⁢ 
                           
                             ≤ 
                             _ 
                           
                           ⁢ 
                           u 
                           ⁢ 
                           
                             ≤ 
                             _ 
                           
                           ⁢ 
                           
                             ( 
                             
                               i 
                               - 
                               2 
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                           
                             n 
                             
                               u 
                               - 
                               
                                 ( 
                                 
                                   i 
                                   - 
                                   j 
                                 
                                 ) 
                               
                             
                           
                           = 
                           
                             S 
                             u 
                             ′ 
                           
                         
                       
                       
                         
                           
                             ( 
                             
                               i 
                               - 
                               j 
                             
                             ) 
                           
                           ⁢ 
                           
                             ≤ 
                             _ 
                           
                           ⁢ 
                           u 
                           ⁢ 
                           
                             ≤ 
                             _ 
                           
                           ⁢ 
                           
                             ( 
                             
                               i 
                               - 
                               2 
                             
                             ) 
                           
                         
                       
                     
                   
                   } 
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     In the formulas (3) and (4), i denotes the code length, j denotes the information length, and v and u denote the syndrome number. Moreover, α denotes a primitive element on a Galois field (2 4 ). 
     If the code length i=15 and the information length j=9 are assumed, the above described formulas (3) and (4) are developed into formulas (5) and (6).  FIG. 3  shows a process of decoding from the code sequence C 0  to C 14  into the address information n 0  to n 8  in the case where it is assumed that there are errors in C 10 , C 11  and C 12 . 
     
       
         
           
             
               
                 
                   [ 
                   
                     Formula 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     5 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             S 
                             0 
                           
                           = 
                           
                             
                               
                                 
                                   C 
                                   0 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     0 
                                   
                                   ) 
                                 
                               
                               14 
                             
                             + 
                             
                               
                                 
                                   C 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     0 
                                   
                                   ) 
                                 
                               
                               13 
                             
                             + 
                             
                               
                                 
                                   C 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     0 
                                   
                                   ) 
                                 
                               
                               12 
                             
                             + 
                             … 
                             + 
                             
                               
                                 
                                   C 
                                   12 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     0 
                                   
                                   ) 
                                 
                               
                               2 
                             
                             + 
                             
                               
                                 
                                   C 
                                   13 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     0 
                                   
                                   ) 
                                 
                               
                               1 
                             
                             + 
                             
                               
                                 
                                   C 
                                   14 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     0 
                                   
                                   ) 
                                 
                               
                               0 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             S 
                             1 
                           
                           = 
                           
                             
                               
                                 
                                   C 
                                   0 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     1 
                                   
                                   ) 
                                 
                               
                               14 
                             
                             + 
                             
                               
                                 
                                   C 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     1 
                                   
                                   ) 
                                 
                               
                               13 
                             
                             + 
                             
                               
                                 
                                   C 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     1 
                                   
                                   ) 
                                 
                               
                               12 
                             
                             + 
                             … 
                             + 
                             
                               
                                 
                                   C 
                                   12 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     1 
                                   
                                   ) 
                                 
                               
                               2 
                             
                             + 
                             
                               
                                 
                                   C 
                                   13 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     1 
                                   
                                   ) 
                                 
                               
                               1 
                             
                             + 
                             
                               
                                 
                                   C 
                                   14 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     1 
                                   
                                   ) 
                                 
                               
                               0 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             S 
                             2 
                           
                           = 
                           
                             
                               
                                 
                                   C 
                                   0 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     2 
                                   
                                   ) 
                                 
                               
                               14 
                             
                             + 
                             
                               
                                 
                                   C 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     2 
                                   
                                   ) 
                                 
                               
                               13 
                             
                             + 
                             
                               
                                 
                                   C 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     2 
                                   
                                   ) 
                                 
                               
                               12 
                             
                             + 
                             … 
                             + 
                             
                               
                                 
                                   C 
                                   12 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     2 
                                   
                                   ) 
                                 
                               
                               2 
                             
                             + 
                             
                               
                                 
                                   C 
                                   13 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     2 
                                   
                                   ) 
                                 
                               
                               1 
                             
                             + 
                             
                               
                                 
                                   C 
                                   14 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     2 
                                   
                                   ) 
                                 
                               
                               0 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             S 
                             3 
                           
                           = 
                           
                             
                               
                                 
                                   C 
                                   0 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     3 
                                   
                                   ) 
                                 
                               
                               14 
                             
                             + 
                             
                               
                                 
                                   C 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     3 
                                   
                                   ) 
                                 
                               
                               13 
                             
                             + 
                             
                               
                                 
                                   C 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     3 
                                   
                                   ) 
                                 
                               
                               12 
                             
                             + 
                             … 
                             + 
                             
                               
                                 
                                   C 
                                   12 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     3 
                                   
                                   ) 
                                 
                               
                               2 
                             
                             + 
                             
                               
                                 
                                   C 
                                   13 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     3 
                                   
                                   ) 
                                 
                               
                               1 
                             
                             + 
                             
                               
                                 
                                   C 
                                   14 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     3 
                                   
                                   ) 
                                 
                               
                               0 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             S 
                             4 
                           
                           = 
                           
                             
                               
                                 
                                   C 
                                   0 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     4 
                                   
                                   ) 
                                 
                               
                               14 
                             
                             + 
                             
                               
                                 
                                   C 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     4 
                                   
                                   ) 
                                 
                               
                               13 
                             
                             + 
                             
                               
                                 
                                   C 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     4 
                                   
                                   ) 
                                 
                               
                               12 
                             
                             + 
                             … 
                             + 
                             
                               
                                 
                                   C 
                                   12 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     4 
                                   
                                   ) 
                                 
                               
                               2 
                             
                             + 
                             
                               
                                 
                                   C 
                                   13 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     4 
                                   
                                   ) 
                                 
                               
                               1 
                             
                             + 
                             
                               
                                 
                                   C 
                                   14 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     4 
                                   
                                   ) 
                                 
                               
                               0 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             S 
                             5 
                           
                           = 
                           
                             
                               
                                 
                                   C 
                                   0 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     5 
                                   
                                   ) 
                                 
                               
                               14 
                             
                             + 
                             
                               
                                 
                                   C 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     5 
                                   
                                   ) 
                                 
                               
                               13 
                             
                             + 
                             
                               
                                 
                                   C 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     5 
                                   
                                   ) 
                                 
                               
                               12 
                             
                             + 
                             … 
                             + 
                             
                               
                                 
                                   C 
                                   12 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     5 
                                   
                                   ) 
                                 
                               
                               2 
                             
                             + 
                             
                               
                                 
                                   C 
                                   13 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     5 
                                   
                                   ) 
                                 
                               
                               1 
                             
                             + 
                             
                               
                                 
                                   C 
                                   14 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     α 
                                     5 
                                   
                                   ) 
                                 
                               
                               0 
                             
                           
                         
                       
                     
                   
                   } 
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
             
               
                 
                   [ 
                   
                     Formula 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     6 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             
                               S 
                               6 
                               ′ 
                             
                             = 
                             
                               
                                 
                                   
                                     C 
                                     0 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       6 
                                     
                                     ) 
                                   
                                 
                                 14 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     1 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       6 
                                     
                                     ) 
                                   
                                 
                                 13 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     2 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       6 
                                     
                                     ) 
                                   
                                 
                                 12 
                               
                               + 
                               … 
                               + 
                               
                                 
                                   
                                     C 
                                     12 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       6 
                                     
                                     ) 
                                   
                                 
                                 2 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     13 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       6 
                                     
                                     ) 
                                   
                                 
                                 1 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     14 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       6 
                                     
                                     ) 
                                   
                                 
                                 0 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               S 
                               7 
                               ′ 
                             
                             = 
                             
                               
                                 
                                   
                                     C 
                                     0 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       7 
                                     
                                     ) 
                                   
                                 
                                 14 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     1 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       7 
                                     
                                     ) 
                                   
                                 
                                 13 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     2 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       7 
                                     
                                     ) 
                                   
                                 
                                 12 
                               
                               + 
                               … 
                               + 
                               
                                 
                                   
                                     C 
                                     12 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       7 
                                     
                                     ) 
                                   
                                 
                                 2 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     13 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       7 
                                     
                                     ) 
                                   
                                 
                                 1 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     14 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       7 
                                     
                                     ) 
                                   
                                 
                                 0 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               S 
                               8 
                               ′ 
                             
                             = 
                             
                               
                                 
                                   
                                     C 
                                     0 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       8 
                                     
                                     ) 
                                   
                                 
                                 14 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     1 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       8 
                                     
                                     ) 
                                   
                                 
                                 13 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     2 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       8 
                                     
                                     ) 
                                   
                                 
                                 12 
                               
                               + 
                               … 
                               + 
                               
                                 
                                   
                                     C 
                                     12 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       8 
                                     
                                     ) 
                                   
                                 
                                 2 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     13 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       8 
                                     
                                     ) 
                                   
                                 
                                 1 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     14 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       8 
                                     
                                     ) 
                                   
                                 
                                 0 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               S 
                               9 
                               ′ 
                             
                             = 
                             
                               
                                 
                                   
                                     C 
                                     0 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       9 
                                     
                                     ) 
                                   
                                 
                                 14 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     1 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       9 
                                     
                                     ) 
                                   
                                 
                                 13 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     2 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       9 
                                     
                                     ) 
                                   
                                 
                                 12 
                               
                               + 
                               … 
                               + 
                               
                                 
                                   
                                     C 
                                     12 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       9 
                                     
                                     ) 
                                   
                                 
                                 2 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     13 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       9 
                                     
                                     ) 
                                   
                                 
                                 1 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     14 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       9 
                                     
                                     ) 
                                   
                                 
                                 0 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               S 
                               10 
                               ′ 
                             
                             = 
                             
                               
                                 
                                   
                                     C 
                                     0 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       10 
                                     
                                     ) 
                                   
                                 
                                 14 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     1 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       10 
                                     
                                     ) 
                                   
                                 
                                 13 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     2 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       10 
                                     
                                     ) 
                                   
                                 
                                 12 
                               
                               + 
                               … 
                               + 
                               
                                 
                                   
                                     C 
                                     12 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       10 
                                     
                                     ) 
                                   
                                 
                                 2 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     13 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       10 
                                     
                                     ) 
                                   
                                 
                                 1 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     14 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       10 
                                     
                                     ) 
                                   
                                 
                                 0 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               S 
                               11 
                               ′ 
                             
                             = 
                             
                               
                                 
                                   
                                     C 
                                     0 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       11 
                                     
                                     ) 
                                   
                                 
                                 14 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     1 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       11 
                                     
                                     ) 
                                   
                                 
                                 13 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     2 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       11 
                                     
                                     ) 
                                   
                                 
                                 12 
                               
                               + 
                               … 
                               + 
                               
                                 
                                   
                                     C 
                                     12 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       11 
                                     
                                     ) 
                                   
                                 
                                 2 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     13 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       11 
                                     
                                     ) 
                                   
                                 
                                 1 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     14 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       11 
                                     
                                     ) 
                                   
                                 
                                 0 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               S 
                               12 
                               ′ 
                             
                             = 
                             
                               
                                 
                                   
                                     C 
                                     0 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       12 
                                     
                                     ) 
                                   
                                 
                                 14 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     1 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       12 
                                     
                                     ) 
                                   
                                 
                                 13 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     2 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       12 
                                     
                                     ) 
                                   
                                 
                                 12 
                               
                               + 
                               … 
                               + 
                               
                                 
                                   
                                     C 
                                     12 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       12 
                                     
                                     ) 
                                   
                                 
                                 2 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     13 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       12 
                                     
                                     ) 
                                   
                                 
                                 1 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     14 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       12 
                                     
                                     ) 
                                   
                                 
                                 0 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               S 
                               13 
                               ′ 
                             
                             = 
                             
                               
                                 
                                   
                                     C 
                                     0 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       13 
                                     
                                     ) 
                                   
                                 
                                 14 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     1 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       13 
                                     
                                     ) 
                                   
                                 
                                 13 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     2 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       13 
                                     
                                     ) 
                                   
                                 
                                 12 
                               
                               + 
                               … 
                               + 
                               
                                 
                                   
                                     C 
                                     12 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       13 
                                     
                                     ) 
                                   
                                 
                                 2 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     13 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       13 
                                     
                                     ) 
                                   
                                 
                                 1 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     14 
                                     ′ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       α 
                                       13 
                                     
                                     ) 
                                   
                                 
                                 0 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               n 
                               0 
                             
                             = 
                             
                               S 
                               6 
                               ′ 
                             
                           
                         
                       
                     
                     } 
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       n 
                       1 
                     
                     = 
                     
                       S 
                       7 
                       ′ 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       n 
                       2 
                     
                     = 
                     
                       S 
                       8 
                       ′ 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       n 
                       3 
                     
                     = 
                     
                       S 
                       9 
                       ′ 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       n 
                       4 
                     
                     = 
                     
                       S 
                       10 
                       ′ 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       n 
                       5 
                     
                     = 
                     
                       S 
                       11 
                       ′ 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       n 
                       6 
                     
                     = 
                     
                       S 
                       12 
                       ′ 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       n 
                       7 
                     
                     = 
                     
                       S 
                       13 
                       ′ 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       n 
                       8 
                     
                     = 
                     
                       C 
                       0 
                       ′ 
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     In an address format described above, in order to decode the address information n 0  to n(j−2) from the code sequence C 0  to C(i−1), the syndrome calculations having two steps, that is, first syndrome calculations S 0  to S(i−j−1) for performing the error detection and correction and second syndrome calculations S(i−j) to S(i−2) for extracting the address information, are required. 
     As a configuration for realizing such two-step syndrome calculations, there is an approach to sequentially processing the two-step syndrome calculations step by step, as shown in Japanese Patent Laid-Open No. 2000-286718. 
     Assuming the code length i=15 and the information length j=9, an approach to decoding the address information n 0  to n(j−2) from the code sequence C 0  to C(i−1) according to the configuration shown in Patent Document 1 will be described. In  FIG. 3 , first, C 0  to C 14  are inputted and first syndromes S 0  to S 5  are calculated as the first step (step A 0 ). 
     Next, according to error determination with S 0  to S 5  (step A 1 ), if it is determined that there is no error in C 0  to C 14  (S 0  to S 5  are all zero), second syndromes S 6  to S 13  are calculated as the second step (step A 6 ), and the address information n 0  to n 7  is outputted. Moreover, C 0  is outputted as n 8 . 
     On the other hand, according to the above described error determination (step A 1 ), if it is determined that there is an error in C 0  to C 14  (at least one of S 0  to S 5  is not zero), S 0  to S 5  are used to perform the error correction with respect to C 0  to C 14 . First, an error position polynomial is derived (step A 2 ), and subsequently, error positions αl, αm and αn and error numbers el, em and en are calculated (steps A 3  and A 4 ), and then a code sequence C′ 0  to C′ 14  in which the errors have been corrected is generated (step A 5 ). 
     Next, these C′ 0  to C′ 14  are used to execute the syndrome calculations at the second step to calculate S 6  to S 13 , and n 0  to n 7  are outputted as estimated values for the address information (step A 6 ). Moreover, C′ 0  is outputted as n 8 . 
       FIG. 4  shows an error correction decoding apparatus applied with a processing scheme according to Patent Document 1, and is a configuration diagram in the case of assuming the code length i=15 and the information length j=9. The error correction decoding apparatus in  FIG. 4  is configured with a first syndrome calculation unit  00  which inputs C 0  to C 14  and calculates S 0  to S 5 , an error position estimation unit  01  which determines the error positions in C 0  to C 14 , an error number estimation unit  02  which determines the error numbers in C 0  to C 14 , an error correction unit  03  which performs the error correction with respect to C 0  to C 14 , and a second syndrome calculation unit  04  which inputs C′ 0  to C′ 14  and calculates S 6  to S 13 . 
     Next, an operation of the error correction decoding apparatus in  FIG. 4  will be described. First, the first syndrome calculation unit  00  inputs C 0  to C 14  and product-sum repeat circuits  000  to  005  calculate S 0  to S 5 . 
     Next, the error position estimation unit  01  derives the error position polynomial from a solution of simultaneous equations of S 0  to S 5 , and estimates the error positions. Moreover, the error number estimation unit  02  estimates the error numbers from the solution of the simultaneous equations of S 0  to S 5 . 
     Next, the error correction unit  03  performs the error correction with respect to C 0  to C 14  based on the above described error positions and error numbers, and generates C′ 0  to C′ 14 . 
     Next, the second syndrome calculation unit  04  inputs C′ 0  to C′ 14 , product-sum repeat circuits  040  to  047  calculate S 6  to S 13 , and its result is outputted as the address information n 0  to n 7 . Moreover, C′ 0  is outputted as n 8 . 
     [Patent Document 1] Japanese Patent Laid-Open No. 2000-286718 
     The above described decoding process is processed in an address decoding process within the recording and playing apparatus for the optical disk in the address format requiring two-step syndrome calculations. 
     When the approach shown in Patent Document 1 is applied in the above described address format, in an error correction decoding method, the first syndrome calculations S 0  to S(i−j−1) for performing the error detection and correction and the second syndrome calculations S(i−j) to S(i−2) for extracting the address information are divided into two steps, and after the first syndrome calculations have been completed and the errors in the code sequence to be used in the second syndrome calculations have been corrected, the above described code sequence C′ 0  to C′ 14  after the error correction is used to perform the second syndrome calculations. Therefore, it is necessary to execute the first syndrome calculations and the second syndrome calculations in this order. 
     Generally, if a product-sum repeat calculation is used to perform the syndrome calculations, product-sum calculations are performed while one symbol of C 0  to C(i−1) is repeatedly inputted per cycle. Therefore, a time required for the syndrome calculations per step is the number of cycles equal to the code length (i cycles). 
     For example, first, in order to perform the error determination, it is necessary to obtain the first syndromes S 0  to S(i−j−1), and the calculation time at this first step requires i cycles. According to a result of the above described error determination, if there is no error in C 0  to C(i−1), it is necessary to obtain the second syndromes S(i−j) to S(i−2) in order to decode the address information, and the calculation time at this second step requires i cycles. As a result, the entire decoding process requires a processing time of 2*i cycles. 
     On the other hand, according to the result of the above described error determination, if there is the error in C 0  to C(i−1), the error correction is performed with respect to C 0  to C(i−1). However, since a time required for the correction varies depending on the number of errors, the time is assumed as e cycles here. Next, it is necessary to use C′ 0  to C′(i−1) to calculate the second syndromes S(i−j) to S(i−2) in order to decode the address information, and the calculation time at this second step requires i cycles. As a result, the entire decoding process requires the processing time of 2*i+e cycles. 
     In this way, in the conventional approach to dividing the syndrome calculations into two steps and processing them as shown in Patent Document 1, the syndrome calculation time occupies a high percentage of the entire decoding process. 
     SUMMARY 
     In accordance with the present invention, there is disclosed an error correction decoding circuit for a playing signal having a code sequence having a decoding unit generating first decoded signal and second decoded signal based on the code sequence and an error correction unit performing error correction for the second signal in response to the first signal. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a diagram showing a method of decoding from a code sequence into address information (in the case where there is no error in the code sequence); 
         FIG. 2  is a diagram showing the method of decoding from the code sequence into the address information (in the case where there is an error in the code sequence); 
         FIG. 3  is a flowchart showing an error correction decoding flow applied with an approach shown in Japanese Patent Laid-Open No. 2000-286718; 
         FIG. 4  is a block diagram showing a configuration example of an error correction decoding apparatus applied with the approach shown in Japanese Patent Laid-Open No. 2000-286718; 
         FIG. 5  is a block diagram showing a whole image of an optical disk recording and playing apparatus; 
         FIG. 6  is a flowchart showing an error correction decoding flow of the present invention; 
         FIG. 7  is a block diagram showing a configuration example of an error correction decoding apparatus of the present invention; 
         FIG. 8  is a timing chart showing a processing time for the error correction decoding apparatus applied with the approach shown in Japanese Patent Laid-Open No. 2000-286718 (in the case where there is no error in the code sequence); 
         FIG. 9  is a timing chart showing a processing time for the error correction decoding apparatus of the present invention (in the case where there is no error in the code sequence); 
         FIG. 10  is a timing chart showing the processing time for the error correction decoding apparatus applied with the approach shown in Japanese Patent Laid-Open No. 2000-286718 (in the case where there is the error in the code sequence); and 
         FIG. 11  is a timing chart showing the processing time for the error correction decoding apparatus of the present invention (in the case where there is the error in the code sequence). 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     This embodiment can reduce a time required for first and second syndrome calculations by parallelly executing the first and second syndrome calculations which have been serially executed. Details thereof will be described below by using the drawings. 
       FIG. 7  shows an error correction decoding apparatus which is applied to the embodiment of the present invention in the case of assuming a code length i=15 and an information length j=9. The error correction decoding apparatus is configured with a syndrome calculation unit  20  which inputs C 0  to C 14  and calculates S 0  to S 13 , an error position estimation unit  21  which determines error positions in C 0  to C 14 , an error number estimation unit  22  which determines error numbers in C 0  to C 14 , an error compensation unit  23  which compensates an amount of errors included in S 6  to S 13  obtained by the syndrome calculation unit  20 , and an C 0  error correction unit  24  which generates n 8 . 
     In this configuration, product-sum repeat circuits  200  to  205  which perform the first syndrome calculations and product-sum repeat circuits  206  to  213  which perform the second syndrome calculations parallelly execute the calculations. Subsequently, based on outputs of the first syndrome calculations (outputs of the product-sum repeat circuits  200  to  205 ), the error position estimation unit  21  outputs signals αl, αm and αn for identifying the error positions. The error number estimation unit  22  receives the outputs of the first syndrome calculations and the output of the error position estimation unit  21 , and outputs signals el, em and en for determining the error numbers by the C 0  error correction unit  24 . The error compensation unit  23  receives outputs of the second syndrome calculations, that is, outputs of the product-sum repeat circuits  206  to  213 , the signals αl, αm and αn for identifying the error positions from the error position estimation unit  21 , and the signals el, em and en for determining the error numbers from the error number estimation unit  22 , and outputs address information n 0  to n 7  after error correction. In this way, the address information in which errors have been compensated can be obtained while the first syndromes and the second syndromes are parallelly calculated, by providing the error compensation unit which receives the outputs of the second syndrome calculations, the output of the error position estimation unit  21  and the output of the error number estimation unit  22  and compensates the errors. 
     Next, an operation of the error correction decoding apparatus in  FIG. 7  will be described. First, the syndrome calculation unit  20  inputs C 0  to C 14 , and the product-sum repeat circuits  200  to  205  and  206  to  213  parallelly calculate first syndromes S 0  to S 5  and second syndromes S 6  to S 13 , respectively. 
     Next, the error position estimation unit  21  derives an error position polynomial from a solution of simultaneous equations of S 0  to S 5  and estimates the error positions. Moreover, the error number estimation unit  22  estimates the error numbers from the solution of the simultaneous equations of S 0  to S 5 . 
     Next, the error compensation unit  23  estimates the amount of errors included in S 6  to S 13  from error information on C 0  to C 14  by using a Galois field adder/multiplier, adds the amount of errors as an compensation value to S 6  to S 13  which have been already obtained, and outputs n 0  to n 7  as the address information. 
     Moreover, in a process of obtaining the above described compensation value, constant tables  230  to  237  are used as circuits for obtaining exponentiation values of the error positions αl, αm and αn, which is intended to further speed up the process. Since objects to be exponentiated are constants (α 6  to α 13 ) and furthermore the number of exponentiations is 0 to 14, which means a small number of patterns, a table configuration is enabled. 
     Moreover, the error-correction unit  24  performs the error correction with respect to C 0  in parallel with the above described compensation process and obtains n 8 . 
     If there is no error in C 0  to C(i−1), a processing time for an error correction decoding method of dividing the syndrome calculations into two steps as shown in Patent Document 1 is (2*i) cycles, whereas a processing time for the present invention is i cycles, which enables a high speed decoding process.  FIG. 8  shows the processing time for the approach of Patent Document 1 and  FIG. 9  shows the processing time for the present invention, in the case where the code length i=15 and the information length j=9 are assumed and there is no error in a code sequence. 
     In a timing chart showing the approach according to Patent Document 1 as shown in  FIG. 8 , when C 0  to C 14  have been inputted as the code sequence, the calculations of S 0  to S 5  are completed at the 15th cycle in the first syndrome calculations, and subsequently, the second syndrome calculations are started from the 15th cycle and the calculations of S 6  to S 13  are completed at the 30th cycle. Therefore, the address information n 0  to n 8  will be obtained after 30 cycles from the input of C 0  which is the beginning of the code sequence. 
     In contrast to this, in a timing chart showing an approach according to the embodiment of the present application as shown in  FIG. 9 , when C 0  to C 14  have been inputted as the code sequence, the first syndrome calculations and the second syndrome calculations are parallelly executed, and thereby S 0  to S 5  and S 6  to S 13  will be obtained at the 15th cycle. Therefore, the address information n 0  to n 8  will be obtained after 15 cycles from the input of C 0  which is the beginning of the code sequence, and the address information can be outputted in a time which is half of that in  FIG. 8 . 
     On the other hand, if there is an error in C 0  to C(i−1), the processing time for the error correction decoding method of dividing the syndrome calculations into two steps as shown in Patent Document 1 is (2*i)+e cycles, whereas the processing time for the present invention is i+e+1 cycles, which enables the high speed decoding process. It should be noted that since the processing time required for error determination and error information generation varies depending on the number of errors, the processing time is assumed as e cycles here.  FIG. 10  shows the processing time for the approach of Patent Document 1 and  FIG. 11  shows the processing time for the present invention, in the case where the code length i=15 and the information length j=9 are assumed and there is the error in the code sequence. 
     A timing chart showing the approach according to Patent Document 1 as shown in  FIG. 10  is similar to  FIG. 8  until the first syndromes are obtained. However, subsequently, the cycle e for the error position and number estimation as well as the error correction is executed, and then the second syndrome calculations are executed similarly to  FIG. 8 . Thereby, the address information is outputted in e+30 cycles. 
     In contrast to this, a timing chart of the embodiment of the present application as shown in  FIG. 11  is the same as  FIG. 9  until the first syndrome calculations and the second syndrome calculations are obtained. However, subsequently, the error position and number estimation as well as the error correction are executed, with which a result of the second syndrome calculations is corrected. Therefore, if the cycle of the error compensation is 1 cycle, the address information can be outputted in e+15+1 cycles. It should be noted that, in the process of obtaining the compensation value, as shown in  FIG. 7 , the exponentiation values of the error positions can be calculated from the constant tables and furthermore the compensation value can be calculated in 1 cycle with simple multiplication and addition. 
     It should be noted that, of course, the embodiment of the present application can be appropriately modified in a range not deviating from the gist of the above described invention.