Abstract:
A decoding method of a product code calculates a kth soft output value of each of r C1 codewords [C t ] (t=1, 2, . . . , r) detected at a codeword generating step. Beginning from t=1, if a kth value of a C1 codeword [C t ] is zero, then compare the first variable, the initial value of which is predetermined, with the likelihood of the codeword, and substitute into the first variable the sum of a greater one of the first variable and the likelihood and a correction value of the difference between them. If the kth value is nonzero, then the second variable is updated in the same manner. The update of the first and second variables is carried out with incrementing t one by one from one to r, and the kth soft output value is calculated from the difference between the first and second variables updated.

Description:
BACKGROUND OF THE INVENTION  
         [0001]    1. Field of the Invention  
           [0002]    The present invention relates to a decoding method and decoding apparatus of a product code for improving the reliability of a digital communication system or digital recording system.  
           [0003]    2. Description of Related Art  
           [0004]    Although error-correcting codes such as Reed-Solomon codes have been utilized to improve reliability of digital communication/recording systems, more powerful error-correcting codes have been required recently to meet increasing speed and capacity of the systems. In general, high performance correcting codes require complicated decoding, which makes it difficult to be built into a device. However, using concatenated codes or product codes makes it possible to implement high-performance coding rather easily. In particular, the product codes have an advantage that they have large redundancy and error-correcting power because of the double encoding of information data. Accordingly, they are applied as an error-correcting method of CD-ROMs and DVDs. FIG. 1 is a diagram showing a construction of a product code. In FIG. 1, a code in the vertical direction is a binary linear code C1 with a code length N1 and information length K1, and a code in the horizontal direction is a binary linear code C2 with a code length N2 and information length K2. The block  1  designates information data, and blocks  2 - 4  each designate a check (redundant check).  
           [0005]    Next, a coding method of the product code will be described with reference to FIG. 1. First, K1×K2 bit-information data are stored in the two-dimensional array (block  1 ) with K1 rows and K2 columns. The block  1  is represented by expression (1), where D i,j  (i=1, 2, . . . , K1; j=1, 2, . . . , K2) is 0 or 1.  
               [           D     1   ,   1             D     1   ,   2           ⋯         D     1   ,   K2                 D     2   ,   1             D     2   ,   2           ⋯         D     2   ,   K2               ⋮       ⋮       ⋰       ⋮             D     K1   ,   1             D     K1   ,   2           ⋯         D     K1   ,   K2             ]                     (     Di   ,     j   =   0     ,   1     )             (   1   )                               
 
           [0006]    Next, individual columns from the first to K2th column are provided with (N1-K1)-bit checks of the C1 code to construct a two-dimensional array with N1-rows and K2-columns (construction of the block  2 ).  
           [0007]    Subsequently, individual rows from the first to N1th row are provided with (N2-K2)-bit checks of the C2 code to construct a two-dimensional array with N1-rows and N2-columns (construction of the blocks  3  and  4 ).  
           [0008]    [0008]FIG. 2 is a block diagram showing a configuration of a digital communication system utilizing a product code, which is disclosed in Japanese patent application laid-open No. 7-202722. In FIG. 2, the reference numeral  201  designates an encoder for encoding input information data;  202  designates a modulator for converting a product code generated by the encoder  201  into a signal suitable for a communication channel;  203  designates the communication channel;  204  designates a demodulator for demodulating a received signal sent via the communication channel and for converting it to demodulated data; and  205  designates a decoder for decoding the demodulated data demodulated by the demodulator to estimate the information data. The encoder  201  and modulator  202  constitute a transmitter, and the demodulator  204  and the decoder  205  constitute a receiver.  
           [0009]    Next, the operation of the system as shown in FIG. 2 will be described. The input K1×K2-bit information data are supplied to the encoder  201  for generating the product code with N1 rows and N2 columns. The product code generated is represented by a matrix C of following Expression 2. Although the elements of the product code C is represented by the binary 0 or 1, the binary 0 is represented as “+1”, and the binary  1  is represented by “−1” in the following.  
             C   =       [           C     1   ,   1             C     1   ,   2           ⋯         C     1   ,   N2                 C     2   ,   1             C     2   ,   2           ⋯         C     2   ,   N2               ⋮       ⋮       ⋰       ⋮             C     N1   ,   1             C     N1   ,   2           ⋯         C     N1   ,   N2             ]                     (       C     i   ,   j       =     ±   1       )               (   2   )                               
 
           [0010]    The product code generated by the encoder  201  is supplied to the modulator  202  to be converted into the signal suitable for the communication channel  203 , and is transmitted through the communication channel. It is assumed about the communication channel  203  that additive noise is superimposed on a transmission signal. The signal received through the communication channel  203  is supplied to the demodulator  204  of the receiver.  
           [0011]    The demodulator  204  carries out shaping of the received signal to generate the demodulated data Y (expression (3)), the components of which are represented as Y i,j =C i,j +N i,j , where N i,j  are noise components. The demodulated data Y generated by the demodulator  204  is supplied to the decoder  205  for estimating the transmitted information data. In the following description, the demodulated data Y is denoted by {Y}, and is referred to as an input matrix.  
             Y   =     [           Y     1   ,   1             Y     1   ,   2           ⋯         Y     1   ,   N2                 Y     2   ,   1             Y     2   ,   2           ⋯         Y     2   ,   N2               ⋮       ⋮       ⋰       ⋮             Y     N1   ,   1             Y     N1   ,   2           ⋯         Y     N1   ,   N2             ]             (   3   )                               
 
           [0012]    [0012]FIG. 3 is a flowchart illustrating the operation of the decoder  205 . In FIG. 3, 301 is a step of inputting the input matrix {Y};  302  is a step of setting initial values into a correction matrix {W} and a decision matrix {D};  303 A is a step of setting an initial value into a counter j;  304 A is a step of calculating soft input vectors [R k ] (k=1, 2, . . . , N1);  305 A is a step of calculating soft output vectors [L k ] (k=1, 2, . . . , N1);  306 A is a step of updating the correction matrix {W};  307 A is a step of comparing the value of the counter j; and  308 A is a step of incrementing the value of the counter j.  
           [0013]    In addition,  303 B is a step of setting an initial value into a counter i;  304 B is a step of calculating soft input vectors [R k ] (k=1, 2, . . . , N2);  305 B is a step of calculating soft output vectors [L k ] (k=1, 2, . . . , N2);  306 B is a step of updating the correction matrix {W};  307 B is a step of comparing the value of the counter i;  308 B is a step of incrementing the value of the counter i;  309  is a step of making a decision as to whether to iterate the decoding of the product code; and  310  is a step of outputting the decision matrix {D}.  
           [0014]    Next, the operation will be described in more detail with reference to the flowchart of FIG. 3. First, at step  301 , the N1×N2 input matrix {Y} given by the foregoing expression (3) is input. At the next step  302 , the initial value zero is stored into all the elements of the N1×N2 correction matrix {W} given by the following expression (4).  
               W   =     [           W     1   ,   1             W     1   ,   2           ⋯         W     1   ,   N2                 W     2   ,   1             W     2   ,   2           ⋯         W     2   ,   N2               ⋮       ⋮       ⋰       ⋮             W     N1   ,   1             W     N1   ,   2           ⋯         W     N1   ,   N2             ]                          (   4   )                               
 
           [0015]    Furthermore, initial values sgn{Y} are stored into all the elements of the N1×N2 decision matrix {D}. Specifically, the (i,j) elements D i,j  of the decision matrix {D} is replaced by the code sgn(Y i,j ) of the (i,j) elements Y i,j  of the input matrix {Y}, where sgn is a function defined by the following expression (5).  
               sgn        (   x   )       =     {             +   1                     (     x   ≥   0     )                   -   1                     (     x   &lt;   0     )                       (   5   )                               
 
           [0016]    At step  303 A, the initial value one is set into the counter j. At the next step  304 A, decoding of the C1 code is started. At step  304 A, jth column of the input matrix {Y} and the jth column of the correction matrix {W} are added element by element. Specifically, according to expression (6), the (k,j) elements Y k,j  of the input matrix {Y} are added to the (k,j) elements W k,j  of the correction matrix {W} to calculate the soft input values R k  (k=1, 2, . . . , N1).  
             R   k   ←Y   k,j   +α·W   k,j  ( k= 1, 2, . . . ,  N 1)  (6)  
           [0017]    where α is an appropriate normalizing constant.  
           [0018]    In the following description, the jth column of the input matrix is denotedby [Y k,j ], that of the decisionmatrix is denoted by [D k,j ], and that of the correction matrix is denoted by [W k,j ], which are called input vector, decision vector and correction vector, respectively, according to the foregoing Japanese patent application laid-open No. 7-202722. At step  305 A, the decision vector [D k,j ] is updated and the soft output vector [L k ] (k=1, 2, . . . , N1) is calculated. The details of step  305 A will be described later. At step  306 A, the differences obtained by subtracting the soft input vector from the soft output vector calculated at step  305 A are stored in the jth column of the correction matrix {W} according to the following expression (7).  
             W   k,j   ←L   k   −R   k  ( k= 1, 2, . . . ,  N 1)  (7)  
           [0019]    At step  307 A, a decision is made as to whether the value of the counter j is less than N2. If it is less than N2, the value of the counter j is incremented at step  308 A, followed by iterating the processing at step  304 A and on. On the other hand, if the value of the counter j is N2, the processing proceeds to step  303 B, at which the decoding of the C2 code is started. Up to this time, the update of all the elements of the correction matrix {W} has been completed.  
           [0020]    At step  303 B, the initial value one is set into the counter i. At the next step  304 B, ith row of the input matrix {Y} and the ith row of the correction matrix {W} are added element by element. Specifically, according to the following expression (8), the (i,k) elements Y i,k  of the input matrix are added to the (i,k) elements W i,k  of the correction matrix to calculate the soft input values R k  (k=1, 2, . . . , N2).  
             R   k   ←Y   i,k   +α·W   i,k  ( k= 1, 2, . . . ,  N 2)  (8)  
           [0021]    where α is an appropriate normalizing constant.  
           [0022]    In the following description, the ith row of the input matrix is denoted by [Y i,k ], that of the decision matrix is denoted by [D i,k ], and that of the correction matrix is denoted by [W i,k ], which are called input vector, decision vector and correction vector, respectively, as in the decoding of the foregoing C1 code. At step  305 B, the decision vector [D i,k ] is updated and the soft output vector [L k ] (k=1, 2, . . . , N2) is calculated. The details of step  305 B will be described later. At step  306 B, the differences obtained by subtracting the soft input vector from the soft output vector calculated at step  305 B are stored in the ith rows of the correction matrix {W} according to the following expression (9).  
             W   i,k   ←L   k   −R   k  ( k= 1, 2, . . . ,  N 2)  (9)  
           [0023]    At step  307 B, a decision is made as to whether the value of the counter i is less than N1. If it is less than N1, the value of the counter i is incremented at step  308 B, followed by iterating the processing at step  304 B and on. On the other hand, if the value of the counter i is N1, the processing proceeds to step  309 . Up to this time, the decoding of the C1 code and C2 code constituting the product codes has been completed once. At step  309 , a decision is made as to whether the decoding of the C1 code is iterated or not. Usually, the decoding is completed when the iterated decoding has been carried out by a predetermined number of times. To iterate the decoding of the C1 code, the processing proceeds to step  303 A to restart the decoding of the C1 code. On the other hand, to stop the decoding, the processing proceeds to step  310 , at which the decision matrix {D} is output. Thus, the decoding processing is completed.  
           [0024]    The data D i,j  (i=1, 2, . . . , K1; j=1, 2, . . . , K2) stored in the K1×K2 decision matrix {D} output at step  310  represent the information data estimated by the decoding. Although the elements of the decision matrix {D} take a value “±1”, the value “+1” corresponds to a binary zero, and “−1” corresponds to a binary one.  
           [0025]    Next, the soft input/soft output decoding of the C1 code at step  305 A will be described. FIG. 4 is a flowchart illustrating the details of step  305 A. Referring to FIG. 4, the operation will be described. At step  401 , the soft input vector [R k ] and the decision vector [D k ] are input.  
           [0026]    At step  402 , p elements with least absolute values are selected from the soft input vector [R]. The positions of the p elements are denoted by k1, k2, . . . , and kp. At step  403 , a test vector [T k ] is generated whose elements T km =0 or 1 at the p positions km (m=1, 2, . . . , p) selected at step  402 , with the remaining elements T k =0 (k≠km). Since the total of q=2 p  test vectors are present, they are denoted as [T s ] (s=1, 2, . . . , q) using the suffix s. The resultant test vectors [T s ] and the decision vector [D k ] are added element by element to generate words [U s ] given by the following expression (10) for carrying out algebraic decoding of the C1 code. In expression (10), the elements “+1” and “−1” of the decision vector [D k ] is converted to a binary zero and one, respectively, and are subjected to modulo-2 addition.  
           [ U   s   ]=[D   k   ]+[T   s ] ( s= 1, 2, . . . ,  q )  (10)  
           [0027]    At step  404 , r candidate codewords [C t ]=(C t   1 , C t   2 , . . . , C t   N1 ) (t=1, 2, . . . , r) are generated by decoding the q words [U s ] (s=1, 2, . . . , q) generated at step  403  using the algebraic decoding of the C1 code.  
           [0028]    At step  405 , Euclidean distances M t  (t=1, 2, . . . , r) are calculated between the soft input vector [R] and the candidate codewords [C t ]. The Euclidean distance M t  between the soft input vector [R] and the candidate codewords [C t ] are given by the following expression (11).  
               M   t     =       ∑     k   =   1     N1            (       R   k     -     C   k   t       )     2               (   11   )                               
 
           [0029]    At step  406 , the codeword [C d ] that gives the minimum Euclidean distance (M t ≧M d ) is selected. In addition, the codewords [C d ] are substituted into the decision vector according to the following expression (12).  
           [D]←[C d ]  (12)  
           [0030]    At step  407 , the counter k is set at its initial value one, and the processing proceeds to step  408 . At step  408 , a decision is made as to whether any candidate codeword [C t ] (t=1, 2, . . . , r) is present, the kth value C t   k  of which differs from the kth value C d   k  of the codeword [C d ] selected at step  406 , that is, whether the codeword [C t ] that satisfies C t   k =−C d   k  is present or not. If it is not present, the processing proceeds to step  409 . If it is present, the processing proceeds to step  410  at which a codeword that gives the least Euclidean distance among such codewords, which is called a concurrent codeword and denoted by [C c ], is selected, and the processing proceeds to step  411 . At step  409 , the soft output value given by the following expression (13) is calculated.  
             L   k   ←βC   d   k   (13)  
           [0031]    where β is an appropriate value.  
           [0032]    At step  411 , the soft output value given by the following expression (14) is calculated.  
             L   k ←(( M   c   −M   d )/4) C   d   k   (14)  
           [0033]    where M c  is a Euclidean distance between the concurrent codeword [C c ] and the soft input vector [R].  
           [0034]    At step  412 , a decision is made as to whether the value of the counter k equals N1 or not. If they are not equal, the processing proceeds to step  413 , at which the value of the counter k is incremented by one to iterate the processing from step  408  and on. On the other hand, if they are equal, the processing proceeds to step  414 , at which the soft output vectors [L k ] and the decision vectors [D k ] are output, followed by completing the entire processing. Thus, the soft input/soft output decoding of the C1 code at step  305 A has been completed. The soft input/soft output decoding of the C2 code at step  305 B is the same as that of the C1 code.  
           [0035]    As seen from the foregoing expressions (13) and (14), the conventional soft output value is calculated using at most two codewords among the candidate codewords generated at step  404 .  
           [0036]    With the foregoing configuration, the conventional decoding method of a product code has a problem of being unable to reflect the information provided by the candidate codewords other than the codeword [C d ] and the concurrent codeword [C s ] closest to the soft input vectors in terms of the Euclidean distance among the many codewords found at the codeword generating step.  
           [0037]    In addition, it has a problem of losing information about the correction vector obtained at the previous decoding in the update of the correction matrix. This is because the new correction vector is obtained by subtracting the soft input vector from the soft output vector calculated by the soft input/soft output decoding.  
         SUMMARY OF THE INVENTION  
         [0038]    The present invention is implemented to solve the foregoing problems. It is therefore an object of the present invention to provide a decoding method and a decoding apparatus of a product code and a digital transmission system using a product code, which are capable of calculating the soft output value by effectively using the codewords obtained at the codeword generating step, and capable of improving the decoding performance by calculating more accurate correction values.  
           [0039]    According to a first aspect of the present invention, there is provided a decoding method of a product code consisting of a binary linear C1 code and C2 code, the decoding method of a product code including the steps of: generating a first soft input value by adding a received value of the C1 code and a predetermined correction value; generating a first soft output value from the first soft input value by carrying out soft input/soft output decoding of the C1 code; updating the correction value by subtracting the received value of the C1 code from the first soft output value; generating a second soft input value by adding a received value of the C2 code and the correction value; generating a second soft output value from the second soft input value by carrying out soft input/soft output decoding of the C2 code; and estimating a transmitted codeword from the soft output value.  
           [0040]    According to a second aspect of the present invention, there is provided a decoding apparatus of a product code consisting of a binary linear C1 code and C2 code, the decoding apparatus of a product code including: an adder for adding a received value and a correction value to generate a soft input value; a soft input/soft output decoder for generating a soft output value from the soft input value; a subtracter for subtracting the received value from the soft output value to generate the correction value; and a decision circuit for deciding decoded bits from the soft output value, wherein the soft input/soft output decoder includes: a first and second Chase decoding circuits for generating a candidate codeword of the C1 code and C2 code from the soft input value; a candidate codeword likelihood calculation circuit for generating a likelihood of the candidate codeword of at least one of the C1 code and C2 code; and a soft output value calculation circuit for generating the soft output value from the likelihood of the candidate codeword.  
           [0041]    The decoding method and apparatus of a product code in accordance with the present invention offer an advantage of being able to generate a more accurate soft output value. In addition, since they use the correction value that reflects the previous decoding results in generating the soft input value, they can markedly improve the decoding performance. 
       
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0042]    [0042]FIG. 1 is a diagram showing a construction of a product code;  
         [0043]    [0043]FIG. 2 is a block diagram showing a configuration of a digital communication system;  
         [0044]    [0044]FIG. 3 is a flowchart illustrating a conventional decoding method of the product code;  
         [0045]    [0045]FIG. 4 is a flowchart illustrating the details of the block  305 A of FIG. 3;  
         [0046]    [0046]FIG. 5 is a flowchart illustrating a decoding method of a product code of an embodiment 1 in accordance with the present invention;  
         [0047]    [0047]FIG. 6 is a flowchart illustrating the details of the block  505 A of FIG. 5;  
         [0048]    [0048]FIG. 7 is a block diagram showing a configuration of a decoding apparatus of a product code of an embodiment 2 in accordance with the present invention;  
         [0049]    [0049]FIG. 8 is a block diagram illustrating the soft input/soft output decoder of FIG. 7; and  
         [0050]    [0050]FIG. 9 is a block diagram showing a configuration of the soft output value calculation circuit of FIG. 8. 
     
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS  
       [0051]    The invention will now be described with reference to the accompanying drawings.  
         [0052]    Embodiment 1  
         [0053]    The decoding method of a product code of an embodiment  1  in accordance with the present invention will be described with reference to the accompanying drawings including the block diagram of the digital communication system as shown in FIG. 2, which is used to describe the prior art.  
         [0054]    [0054]FIG. 5 is a flowchart illustrating the decoding method of a product code in accordance with the present invention. In FIG. 5, 501 is a step of inputting the input matrix {Y};  502  is a step of setting an initial value into the correction matrix {W};  503 A is a step of setting an initial value into a counter j;  504 A is a step of calculating soft input vectors [R k ] (k=1, 2, . . . , N1);  505 A is a step of calculating soft output vectors [L k ] (k=1, 2, . . . , N1);  506 A is a step of updating the correction matrix {W};  507 A is a step of comparing the value of the counter j; and  508 A is a step of incrementing the value of the counter j.  
         [0055]    In addition,  503 B is a step of setting an initial value into a counter i;  504 B is a step of calculating soft input vectors [R k ] (k=1, 2, . . . , N2);  505 B is a step of calculating soft output vectors [L k ] (k=1, 2, . . . , N2);  506 B is a step of updating the correction matrix {W} and the decision matrix {D};  507 B is a step of comparing the value of the counter i;  508 B is a step of incrementing the value of the counter i;  509  is a step of making a decision as to whether to iterate the decoding of the product code; and  510  is a step of outputting the decision matrix {D}.  
         [0056]    Next, the operation will be described in more detail with reference to the flowchart of FIG. 5. First, at step  501 , the N1×N2 input matrix {Y} given by the foregoing expression (3) is input. At the next step  502 , the initial value zero is stored into all the elements of the N1×N2 correction matrix {W} given by the following expression (15).  
               W   =     [           W     1   ,   1             W     1   ,   2           ⋯         W     1   ,   N2                 W     2   ,   1             W     2   ,   2           ⋯         W     2   ,   N2               ⋮       ⋮       ⋰       ⋮             W     N1   ,   1             W     N1   ,   2           ⋯         W     N1   ,   N2             ]                          (   15   )                               
 
         [0057]    At step  503 A, the initial value one is set into the counter j. At the next step  504 A, decoding of the C1 code is started. At step  504 A, jth column of the input matrix {Y} and the jth column of the correction matrix {W} are added element by element. Specifically, according to the following expression (16), the (k,j) elements Y k,j  of the input matrix are added to the (k,j) elements W k,j  of the correction matrix to calculate the soft input values R k  (k=1, 2, . . . , N1)  
           R   k   ←Y   k,j   +α·W   k,j  ( k= 1, 2, . . . ,  N 1)  (16)  
         [0058]    where α is an appropriate normalizing constant.  
         [0059]    As in the description of the prior art, the jth column of the input matrix is denoted by [Y k,j ], and that of the correction matrix is denoted by [W k,j ] which are called an input vector and a correction vector, respectively. At step  505 A, the soft output vectors [L k ] (k=1, 2, . . . , N1) are calculated from the soft input vectors [R k ] calculated at step  504 A. The details of the calculation method of the soft output vector at step  505 A will be described later. At step  506 A, the differences obtained by subtracting the input values Y k,j  from the soft output values L k  calculated at step  505 A are stored in the jth column of the correction matrix {W} as the following expression (17).  
           W   k,j   ←L   k   −Y   k,j  ( k= 1, 2, . . . ,  N 1)  (17)  
         [0060]    Although the correction vector [W k,j ] is updated by subtracting the soft input vector [R k ] from the soft output vector [L k ] in the prior art (step  306 A of FIG. 3), it is obtained by subtracting the input vector [Y k,j ] from the soft output vector [L k ] in the present invention. Thus, it offers an advantage of being able to obtain more accurate correction vectors better reflecting the previous correction information in the iteration of the decoding.  
         [0061]    At step  507 A, a decision is made as to whether the value of the counter j is less than N2. If it is less than N2, the value of the counter j is incremented at step  508 A, followed by iterating the processing from step  504 A and on. On the other hand, if the value of the counter is N2, the processing proceeds to step  503 B, at which the decoding of the C2 code is started.  
         [0062]    At step  503 B, the initial value one is set into the counter i. At the next step  504 B, the ith row of the input matrix {Y} and the ith row of the correction matrix {W} are added element by element. Specifically, according to the following expression (18), the (i,k) elements Y i,k  of the input matrix are added to the (i,k) elements W i,k  of the correction matrix to calculate the soft input values R k  (k=1, 2, . . . , N2).  
           R   k   ←Y   i,k   +α·W   i,k  ( k= 1, 2, . . . ,  N 2)  (18)  
         [0063]    where α is an appropriate normalizing constant.  
         [0064]    In the following description, the ith row of the input matrix is denoted by [Y i,k ] and that of the correction matrix is denoted by [W i,k ], which are called an input vector and a correction vector, respectively, as in the decoding of the foregoing C1 code. At step  505 B, the soft output vector [L k ] (k=1, 2, . . . , N2) is calculated from the soft input vector [R k ] calculated at step  504 B.  
         [0065]    At step  506 B, the differences obtained by subtracting the input vector from the soft output vector calculated at step  505 B are stored in the ith row of the correction matrix {W} according to the following expression (19).  
           W   i,k   ←L   k   −Y   i,k  ( k= 1, 2, . . . ,  N 2)  (19)  
         [0066]    In addition, the hard decisions of Lk are substituted into the (i,k) elements D i,k  of the decision matrix {D} as shown by the following expression (20).  
               D     l   ,   k       =     {           0         (       L   k     ≥   0     )             1         (       L   k     &lt;   0     )                           (       k   =   1     ,   2   ,   …              ,   N2     )                 (   20   )                               
 
         [0067]    At step  507 B, a decision is made as to whether the value of the counter i is less than N1 or not. If it is less than N1, the value of the counter i is incremented at step  508 B, followed by iterating the processing from step  504 B and on. On the other hand, if the value of the counter i is N1, the processing proceeds to step  509 . Up to this time, the decoding of the C1 code and C2 code constituting the product code has been completed once. At step  509 , a decision is made as to whether to iterate the decoding or not. To iterate the decoding, the processing proceeds to step  503 A to restart the decoding of the C1 code. On the other hand, to stop the decoding, the processing proceeds to step  510 , at which the decision matrix {D} is output. Thus, the processing is completed.  
         [0068]    The data D i,j  (i=1, 2, . . . , K1; j=1, 2, . . . , K2) in the K1×K2 decision matrix {D} represent the estimated information data. The step  509  can calculate the syndromes of the decision matrix {D} to check whether any error remains or not so that if a decision is made that an error is present, the decoding is iterated, or else the decoding can be completed.  
         [0069]    Next, the soft input/soft output decoding of the C1 code at step  505 A will be described. FIG. 6 is a flowchart illustrating the details of step  505 A. Referring to FIG. 6, the operation of the soft input/soft output decoding of the C1 code will be described. At step  601 , the soft input vector [R k ] is input. At step  602 , a hard decision vector [H k ] of the soft input vector [R k ] is generated according to the following expression (21).  
               H   k     =     {         0         (       R   k     ≥   0     )             1         (       R   k     &lt;   0     )                     (   21   )                               
 
         [0070]    At step  603 , p elements with least absolute values are selected from the soft input vector [R k ]. The positions of the p elements selected are denoted by k1, k2, . . . , and kp. At step  604 , a test vector [T] is generated whose elements T km =0 or 1 at the p positions km (m=1, 2, . . . , p) which are selected at step  603 , with the remaining elements T k =0 (k≠km). Since the total of q=2 p  test vectors are present, they are denoted as [T s ] (s=1, 2, . . . , q) using the suffix s. The resultant test vectors [T s ] and the hard decision vector H generated at step  602  are added element by element to generate words [U s ] given by the following expression (22) for carrying out algebraic decoding of the C1 code.  
         [ U   s   ]=[H]+[T   s ] ( s= 1, 2, . . . ,  q )  (22)  
         [0071]    At step  605 , r candidate codewords [C t ]=(C t   1 , C t   2 , . . . , C t   N1 ) (t=1, 2, . . . , r) are generated by decoding the words [U s ], which are generated at step  604 , using the algebraic decoding of the C1 code.  
         [0072]    At step  606 , the inner products P t  (t=1, 2, . . . , r) between the soft input vector [R] and the candidate codewords [C t ] are calculated. The inner products P t between the soft input vector [R] and the candidate codewords [C t ] are given by the following expression (23).  
               P   t     =     2          ∑     k   =   1     N1            R   k          C   k   t                   (   23   )                               
 
         [0073]    At step  607 , the codeword [C d ] that gives the maximum value of the inner products P t  calculated at step  606  is selected. At step  608 , the initial value one is set into the counter k. At step  609 , the initial value one is set in the counter t, and variables L0 and L1 are set at “−∞” that indicates the minimum value the computer or hardware can represent. At step  610 , a decision is made as to whether the kth element C t   k  of the tth candidate codeword [C t ] is zero or not. If the element C t   k  is zero, the processing proceeds to step  611 , or else it proceeds to step  612 .  
         [0074]    At step  611 , the following expression (24) is calculated, and then the processing proceeds to step  613 .  
           L 0 ←f ( L 0 , P   t )  (24)  
         [0075]    At step  612 , the following expression (25) is calculated, and then the processing proceeds to step  613 .  
           L 1 ←f ( L 1 , P   t )  (25)  
         [0076]    where the function f in the expressions (24) and (25) is given by the following expression (26).  
           f ( a,b )=max( a,b )+ log (1+ e   −|a−b| )  (26)  
         [0077]    where “max” indicates to select the greater one of the two variables.  
         [0078]    At step  613 , a decision is made as to whether the counter t is less than the total number r of the candidate codewords generated at step  605  or not. When t is less than r, the processing proceeds to step  614 , at which the value of the counter t is incremented to iterate the processing from step  610  and on. On the other hand, if the counter t agrees with r, the processing proceeds to step  615 .  
         [0079]    At step  615 , a decision is made as to whether the variable L0 or L1 equals −∞ or not. If it is equal, the processing proceeds to step  616 , or else it proceeds to step  617 . At step  616 , the soft output value given by the following expression (27) is calculated.  
           L   k   ←R   k   +βC   d   k   (27)  
         [0080]    where β is an appropriate normalizing constant. At step  617 , the soft output value given by the following expression (28) is calculated.  
           L   k ←( L 0 −L 1)/4  (28)  
         [0081]    At step  618 , a decision is made as to whether the value of the counter k is equal to N1 or not. If k is less than N1, the processing proceeds to step  619 , at which the value of the counter k is incremented to iterate the processing from step  609  and on. On the other hand, if they are equal, the processing proceeds to step  620 , at which the soft output vector [L k ] is output, followed by completing the entire processing. Thus, the soft input/soft output decoding of the C1 code at step  505 A has been completed. The soft input/soft output decoding of the C2 code at step  505 B is the same as that of the C1 code at step  505 A.  
         [0082]    As described above, the present embodiment 1 is configured such that the soft output values L k  are calculated using all the candidate codewords generated at step  605 . Accordingly, it has an advantage of being able to achieve more accurate soft output values than the conventional techniques. In the following paragraphs, the formulae for calculating the soft output values of the prior art and the present invention will be compared. Here, the description is made by way of example of calculating the soft output values of the C1 code. It is known that when the soft input vector [R k ] is given, the accurate value of the kth soft output value L k  is calculated by the following expression (29).  
               L   j     ∝     log                       ∑       C   j     =     +   1                   -       (     R   -   C     )     2               ∑       C   j     =     -   1                   -       (     R   -   C     )     2                       (   29   )                               
 
         [0083]    where the numerator is the sum total of the codewords C=[C k ] (k=1, 2, . . . , N1) of the C1 code having the value C j =+1, and the denominator is the sum total of the codewords C=[C k ] of the C1 code having the C j =−1.  
         [0084]    Assume that the codeword among the codewords C=[C k ], which has the value C j =+1 and is closest to the vector [R k ] in terms of the Euclidean distance is denoted by [C +1   k ], and that the codeword among the codewords C=[C k ], which has the value C j =−1 and is closest to the vector [R k ] in terms of the Euclidean distance is denoted by [C −1   k ], the foregoing expression (29) is approximated by the following expression (30).  
               L   j     =     -           (     R   -     C     +   1         )     2     -       (     R   -     C     -   1         )     2       4               (   30   )                               
 
         [0085]    The calculation of the soft output values of the prior art is based on expression (30). In contrast, the present embodiment, limiting the codewords [C k ] of the denominator and numerator of expression (29) to the candidate codewords [C t ] (t=1, 2, . . . , r) generated at the codeword generating step, calculates the soft output values by the following expression (31) using instead of the Euclidean distance the inner product between the vector [R k ] and the candidate codeword [C t ], which is easier to calculate than the Euclidean distance.  
               L   j     =       1   4          [       log   (       ∑     C     j   =     +   1       t                 2        R   ·     C                t               )     -     log   (       ∑     C     j   =     -   1       t                 2        R   ·     C                t               )       ]               (   31   )                               
 
         [0086]    The first and second terms of the expression (31) can be recursively calculated using the relationship of the following expression (32). In particular, storing quantized values of the function L given by the following expression (33) into a table enables the speedup of the calculation of expression (31).  
           log ( e   a   +e   b )=max( a,b )+ log (1+ e   |a−b| )  (32)  
           L ( x )= log (1+ e   −x ) ( x&gt; 0)  (33)  
         [0087]    With the foregoing configuration, the present embodiment 1 of the decoding method of a product code offers an advantage of being able to generate the more accurate soft output values. In addition, since it is configured such that the correction vectors take account of the previous decoding results, it can improve the decoding performance markedly.  
         [0088]    The present embodiment 1 can be modified in various ways. For example, when the received values of the product codes are given by the hard decision, the decoding method of the present embodiment is applicable under the condition that the soft input values are placed at M when the hard decision is zero, and at −M when the hard decision is one, where M is an appropriate value.  
         [0089]    In addition, it is possible to modify the present embodiment 1 in such a manner that it calculates the syndromes of the product codes at step  509 , and if all the syndromes are zero, it completes the decoding, or else it iterates the decoding. Thus, it offers an advantage of being able to eliminate useless iteration.  
         [0090]    Embodiment 2  
         [0091]    The decoding method of a product code described above in connection with the embodiment 1 can be implemented by hardware. FIG. 7 is a block diagram showing a configuration of a decoding apparatus for the product code including the same C1 code and C2 code. In FIG. 7, the reference numeral  701  designates an adder;  702  designates a subtracter;  703  designates a first memory for storing a received word of the product code supplied from the modulator on the transmitting side;  704  designates a soft input/soft output decoder for carrying out the soft input/soft output decoding of the C1 code and C2 code;  705  designates decision circuit for deciding the transmitted codeword from the soft output values supplied by the soft input/soft output decoder  704 ;  706  designates a third memory for storing the transmitted codeword decided by the decision circuit  705 ;  707  designates a second memory for storing the correction value supplied from the subtracter  702 ; and  708  designates a normalizing circuit for normalizing the correction value supplied from the second memory  707 . In the decision circuit  705 , the reference numeral  705   a  designates a syndrome calculation circuit for calculating the syndromes of the transmitted codeword to decide whether an error is present or not; and  705   b  is an error number calculation circuit for measuring the number of errors from the estimated transmitted codeword and the received word to monitor the state of the communication channel.  
         [0092]    Next, the operation of the decoding apparatus of FIG. 7 will be described. First, the received word Y supplied from the demodulator is stored in the first memory  703 . To decode the C1 code (vertical direction) or C2 code (horizontal direction) of the product code, the correction value stored in the specified address of the second memory  707  is read out and supplied to the adder  701  after passing through the normalization by the normalizing circuit  708 . The adder  701  adds the received value stored at the specified address of the first memory  703  and the correction value supplied from the normalizing circuit  708  to generate the soft input value. At the first decoding, the reading from the second memory  707  is skipped, so that the adder  701  delivers the received value to the soft input/soft output decoder  704  without change.  
         [0093]    The soft input value generated by the adder  701  is supplied to the soft input/soft output decoder  704 . Receiving the soft input value associated with one codeword of the C1 code or C2 code, (that is, the soft input vector described in the foregoing embodiment 1), the soft input/soft output decoder  704  starts the decoding in accordance with the flowchart of FIG. 6.  
         [0094]    [0094]FIG. 8 is a block diagram showing a configuration of the soft input/soft output decoder  704 . In FIG. 8, the reference numeral  801  designates a Chase decoding circuit for generating a candidate of the transmitted codeword from the soft input vector;  802  designates a candidate codeword likelihood calculation circuit for calculating the likelihood of the candidate codeword; and  803  designates a soft output value calculation circuit for calculating the soft output value from the candidate codeword. Since the Chase decoding circuit  801  belongs to a known technique, its details will be omitted here. See, D. Chase, “A class of algorithms for decoding block codes with channel measurement information”, (IEEE Trans. Inform. Theory, Vol. IT-18, pp. 170-182).  
         [0095]    Next, the operation of the soft input/soft output decoder  704  of FIG. 8 will be described. The Chase decoding circuit  801  carries out the processing from step  601  to step  605  of FIG. 6 to generate the candidate codewords [C t ] (t=1, 2, . . . , r) from the soft input vector [R].  
         [0096]    The candidate codewords [C t ] (t=1, 2, . . . , r) generated by the Chase decoding circuit  801  are supplied to the candidate codeword likelihood calculation circuit  802 . The candidate codeword likelihood calculation circuit  802  calculates the inner products P t  between the soft input vector [R] and the candidate codewords [C t ] (t=1, 2, . . . , r) given by the foregoing expression (23), and detects the candidate codeword [C d ] giving the maximum inner product. The candidate codeword likelihood calculation circuit  802  supplies the inner products Pt and the maximum likelihood codeword [C d ] to the soft output value calculation circuit  803 .  
         [0097]    The soft output value calculation circuit  803  generates the soft output values according to the foregoing expression (27) or (28). FIG. 9 shows a configuration of the soft output value calculation circuit  803  for calculating the variable L0 or L1. In FIG. 9, the reference numeral  901  designates an input terminal to which the inner product P t  is applied;  902  designates a register for storing a calculation result;  903  designates a MAX circuit for selecting a greater one of the two inputs and for outputting the selected one;  904  designates a MIN circuit for selecting a smaller one of the two inputs and for outputting the selected one;  905  designates a lookup table that stores the quantized values of the foregoing expression (33);  906  designates a subtracter for subtracting the output of the MIN circuit  904  from the output of the MAX circuit  903 ;  907  designates an adder for adding the output of the MAX circuit  903  and the output of the lookup table  905 ; and  908  designates an output terminal for outputting the content of the register  902 .  
         [0098]    Next, the operation of the soft output value calculation circuit  803  of FIG. 9 will be described by way of example of calculating the variable L0. A sufficiently small value is placed into the register  902  as its initial value. The inner product P t  of the candidate codeword [C t ] whose kth value C t   k  is zero is input from the input terminal  901 . The MAX circuit  903  compares the inner product P t  with the data stored in the register  902 , and selects and supplies the greater one to the subtracter  906  and adder  907 . On the other hand, the MIN circuit  904  compares the inner product P t  with the data stored in the register  902 , and selects and supplies the smaller one to the subtracter  906 .  
         [0099]    The subtracter  906  subtracts the output data of the MIN circuit  904  from the output data of the MAX circuit  903 , and supplies the resultant value to the lookup table  905 . The lookup table  905  reads the logarithmic value given by expression (33), and supplies the resultant value to the adder  907 . The adder  907  adds the output of the MAX circuit  903  and the output of the lookup table  905 , and stores the sum to the register  902 .  
         [0100]    Since the variable L1 can be calculated in the same manner as the variable L0 using the circuit as shown in FIG. 9, the description thereof is omitted here. The soft output value is generated from the calculated variables L0 and L1 using the foregoing expression (28). If the variable L0 or L1 is not calculated, the soft output value is generated from the soft input value and the element of the maximum likelihood codeword [C d ] using expression (27).  
         [0101]    The soft output value generated by the soft output value calculation circuit  803  of the soft input/soft output decoder  704  is supplied to the decision circuit  705  and subtracter  702 . The subtracter  702  subtracts the received value from the soft output value to generate the correction value, and stores it to a particular address of the second memory  707 . The decision circuit  705  makes a decision of the transmitted codeword from the soft output value, and stores it to a particular address of the third memory  706 .  
         [0102]    With the foregoing configuration, the decoding apparatus of a product code of the present embodiment 2 can generate highly accurate soft output values using the lookup table. In addition, since the correction value is generated by subtracting the received value from the soft output value, the present embodiment 2 can generate more effective correction value.  
         [0103]    Although it is assumed that the C1 code is the same as the C2 code in the foregoing description, the decoding apparatus can be configured in the same manner when they are different. Specifically, it is enough for the soft input/soft output decoder to have a first Chase decoder for the C1 code and a second Chase decoder for the C2 code, with sharing the remaining circuits.  
         [0104]    Furthermore, although the foregoing configuration comprises only one soft input/soft output decoder, multiple decoders installed in parallel can enhance the speed of the decoding apparatus.  
         [0105]    Moreover, using the syndrome calculation circuit  705   a , the decision circuit  705  can calculate the syndromes of the transmitted codeword to decide whether an error is present or not. Thus, the present embodiment 2 can iterate the decoding if the error is present, and complete the decoding otherwise. As a result, it offers an advantage of being able to eliminate useless iteration. Furthermore, using the error number calculation circuit  705   b , the decision circuit  705  can measure the number of errors from the estimated transmitted codeword and the received word to monitor the state of the communication channel. This offers an advantage of being able to facilitate the setting of the number of times of the iteration of the decoding. That is, when the state of the communication channel is bad, the number of times of the iteration is increased, and when it is good, it is reduced.  
         [0106]    The decoding apparatus of a product code is suitable for establishing the good digital transmission, and can implement the high performance digital transmission system by connecting the encoding apparatus of the product code with the decoding apparatus via a transmission medium. Incidentally, it is obvious that not only a wireless channel or an optical fiber can be used as the transmission medium, but also a recording medium such as an optical disk is usable.