golay.c

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/* File:    golay.c
 * Title:   Encoder/decoder for a binary (23,12,7) Golay code
 * Author:  Max Luttrell (luttrell@icsl.ucla.edu),
 * Date:    October 1997
 *          
 * based on work done by:
 * Author:  Robert Morelos-Zaragoza (robert@spectra.eng.hawaii.edu)
 * Date:    August 1994
 *
 * The binary (23,12,7) Golay code is an example of a perfect code, that is,
 * the number of syndromes equals the number of correctable error patterns.
 * The minimum distance is 7, so all error patterns of Hamming weight up to
 * 3 can be corrected. The total number of these error patterns is:
 *
 *       Number of errors         Number of patterns
 *       ----------------         ------------------
 *              0                         1
 *              1                        23
 *              2                       253
 *              3                      1771
 *                                     ----
 *    Total number of error patterns = 2048 = 2^{11} = number of syndromes
 *                                               --
 *                number of redundant bits -------^
 *
 * Because of its relatively low length (23), dimension (12) and number of
 * redundant bits (11), the binary (23,12,7) Golay code can be encoded and
 * decoded simply by using look-up tables. The program below uses a 16K 
 * encoding table and an 8K decoding table.
 * 
 * For more information, suggestions, or other ideas on implementing error
 * correcting codes, please contact me at (I'm temporarily in Japan, but
 * below is my U.S. address):
 *
 *                    Robert Morelos-Zaragoza
 *                    770 S. Post Oak Ln. #200
 *                      Houston, Texas 77056
 *
 *             email: robert@spectra.eng.hawaii.edu
 *
 * NOTE: This program is free. You may modify it at will. However, please
 *       include the original file with any redistribution. The author is
 *       not responsible for any damage caused by the use of this program.
 *
 *       Homework: Add an overall parity-check bit to get the (24,12,8)
 *                 extended Golay code.
 *
 * ==   Copyright (c) 1994  Robert Morelos-Zaragoza. All rights reserved.   ==
 */

#include <stdio.h>
#define X22             0x00400000   /* vector representation of X^{22} */
#define X11             0x00000800   /* vector representation of X^{11} */
#define MASK12          0xfffff800   /* auxiliary vector for testing */
#define GENPOL          0x00000c75   /* generator polinomial, g(x) */

/* Global variables:
 *
 * pattern = error pattern, or information, or received vector
 * encoding_table[] = encoding table
 * decoding_table[] = decoding table
 * data = information bits, i(x)
 * codeword = code bits = x^{11}i(x) + (x^{11}i(x) mod g(x))
 * numerr = number of errors = Hamming weight of error polynomial e(x)
 * position[] = error positions in the vector representation of e(x)
 * recd = representation of corrupted received polynomial r(x) = c(x) + e(x)
 * decerror = number of decoding errors
 * a[] = auxiliary array to generate correctable error patterns
 */

long pattern;
long encoding_table[4096], decoding_table[2048];
long data, codeword, recd;
long position[23] = { 0x00000001, 0x00000002, 0x00000004, 0x00000008,
                      0x00000010, 0x00000020, 0x00000040, 0x00000080,
                      0x00000100, 0x00000200, 0x00000400, 0x00000800,
                      0x00001000, 0x00002000, 0x00004000, 0x00008000,
                      0x00010000, 0x00020000, 0x00040000, 0x00080000,
                      0x00100000, 0x00200000, 0x00400000 };
long numerr, errpos[23], decerror = 0;
int a[4];
extern int debug;

long arr2int(int* a,int r)
/*
 * Convert a binary vector of Hamming weight r, and nonzero positions in
 * array a[1]...a[r], to a long integer \sum_{i=1}^r 2^{a[i]-1}.
 */
{
   int i;
   long mul, result = 0, temp;
 
   for (i=1; i<=r; i++) {
      mul = 1;
      temp = a[i]-1;
      while (temp--)
         mul = mul << 1;
      result += mul;
      }
   return(result);
}

void nextcomb(int n, int r, int *a)
/*
 * Calculate next r-combination of an n-set.
 */
{
  int  i, j;
 
  a[r]++;
  if (a[r] <= n)
      return;
  j = r - 1;
  while (a[j] == n - r + j)
     j--;
  for (i = r; i >= j; i--)
      a[i] = a[j] + i - j + 1;
  return;
}

long get_syndrome(long pattern)
/*
 * Compute the syndrome corresponding to the given pattern, i.e., the
 * remainder after dividing the pattern (when considering it as the vector
 * representation of a polynomial) by the generator polynomial, GENPOL.
 * In the program this pattern has several meanings: (1) pattern = infomation
 * bits, when constructing the encoding table; (2) pattern = error pattern,
 * when constructing the decoding table; and (3) pattern = received vector, to
 * obtain its syndrome in decoding.
 */
{
    long aux = X22;
 
    if (pattern >= X11)
       while (pattern & MASK12) {
           while (!(aux & pattern))
              aux = aux >> 1;
           pattern ^= (aux/X11) * GENPOL;
           }
    return(pattern);
}

void gen_enc_table(void)
{
   /*
    * ---------------------------------------------------------------------
    *                  Generate ENCODING TABLE
    *
    * An entry to the table is an information vector, a 32-bit integer,
    * whose 12 least significant positions are the information bits. The
    * resulting value is a codeword in the (23,12,7) Golay code: A 32-bit
    * integer whose 23 least significant bits are coded bits: Of these, the
    * 12 most significant bits are information bits and the 11 least
    * significant bits are redundant bits (systematic encoding).
    * --------------------------------------------------------------------- 
    */
   long temp;
   for (pattern = 0; pattern < 4096; pattern++) {
       temp = pattern << 11;          /* multiply information by X^{11} */
       encoding_table[pattern] = temp + get_syndrome(temp);/* add redundancy */
   }
}

void gen_dec_table(void)
{
   /*
    * ---------------------------------------------------------------------
    *                  Generate DECODING TABLE
    *
    * An entry to the decoding table is a syndrome and the resulting value
    * is the most likely error pattern. First an error pattern is generated.
    * Then its syndrome is calculated and used as a pointer to the table
    * where the error pattern value is stored.
    * --------------------------------------------------------------------- 
    *            
    * (1) Error patterns of WEIGHT 1 (SINGLE ERRORS)
    */
    long temp;
    int i;

    decoding_table[0] = 0;
    decoding_table[1] = 1;
    temp = 1; 
    for (i=2; i<= 23; i++) {
        temp *= 2;
        decoding_table[get_syndrome(temp)] = temp;
        }
   /*            
    * (2) Error patterns of WEIGHT 2 (DOUBLE ERRORS)
    */
    a[1] = 1; a[2] = 2;
    temp = arr2int(a,2);
    decoding_table[get_syndrome(temp)] = temp;
    for (i=1; i<253; i++) {
        nextcomb(23,2,a);
        temp = arr2int(a,2);
        decoding_table[get_syndrome(temp)] = temp;
        }
   /*            
    * (3) Error patterns of WEIGHT 3 (TRIPLE ERRORS)
    */
    a[1] = 1; a[2] = 2; a[3] = 3;
    temp = arr2int(a,3);
    decoding_table[get_syndrome(temp)] = temp;
    for (i=1; i<1771; i++) {
        nextcomb(23,3,a);
        temp = arr2int(a,3);
        decoding_table[get_syndrome(temp)] = temp;
        }
}


long encode_golay(long data)
{
/*
 * encodes data and returns the codeword.  
 * data is assumed to contain 12 bits in the least significant bit positions
 * codeword contains the 23 bits in the LSB positions
 * 
 */
    static int initialized=0;

    if (!initialized) {
        gen_enc_table();
        initialized=1;
    }
    codeword = encoding_table[data];
    if (debug) printf("encode_golay() - codeword =        %#012x\n", codeword);
    return codeword;
}

long decode_golay(long codeword)
{
/*
 * decodes codeword and returns the dataword contained within.  
 * 
 * note: no detection is done here!
 * codeword is assumed to contain the 23 bits in the LSB positions
 * the returned bits contain the bits in the LSB positions
 * 
 * MSB will be ignored, since this is 23,12 code.
 */
   static int initialized=0;

   if (!initialized) {
       gen_dec_table();
       initialized=1;
   }

   if (debug) {
       printf("decode_golay() - codeword =  %#012x\n", codeword);
       printf("decode_golay() - syndrome =  %#012x\n", get_syndrome(codeword));
   }

   /*
    * Calculate the syndrome, look up the corresponding error pattern and
    * add it to the received vector.
    */
   codeword ^= decoding_table[get_syndrome(codeword)];
   
   if (debug) {
     printf("decoded vector =  %#012x\n", codeword);
     printf("recovered data =  %#012x\n", (codeword>>11));
   }
   return codeword >> 11;
}

#if(0)
main()
{
  long data,encoded,decoded;
  data = 0x0b;
  printf("data: %x \n",data);
  encoded = encode_golay(data);
  printf("encoded: %x \n",encoded);
  encoded ^= 0xE000;
  printf("error encoded: %x \n",encoded);
  decoded = decode_golay(encoded);
  printf("decoded: %x \n",decoded);

}
#endif

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