bch_erasures.c

error correction code

// ------------------------------------------------------------------------
//
//        File: bch_erasures.c
//        Date: April 3, 2002
//
// Description: Encoding and decoding routines for binary BCH codes
//
//              Erasure correction by two errors-only decoding passes using 
//              the EUCLIDEAN ALGORITHM
//
// ------------------------------------------------------------------------
// This program is complementary material for the book:
//
// R.H. Morelos-Zaragoza, The Art of Error Correcting Coding, Wiley, 2002.
//
// ISBN 0471 49581 6
//
// This and other programs are available at http://the-art-of-ecc.com
//
// You may use this program for academic and personal purposes only. 
// If this program is used to perform simulations whose results are 
// published in a journal or book, please refer to the book above.
//
// The use of this program in a commercial product requires explicit 
// written permission from the author. The author is not responsible or 
// liable for damage or loss that may be caused by the use of this program. 
//
// Copyright (c) 2002. Robert H. Morelos-Zaragoza. All rights reserved.
// ------------------------------------------------------------------------

#include <math.h>
#include <stdio.h>

int             m, n, length, k, t, d;
int             p[21];
int             alpha_to[  32768], index_of[  32768], g[ 32768];
int             recd[  32768], data[  32768], bb[ 32768];
int             seed;
int             numera, erapos[1024];
int             numerr, errpos[1024], decerror = 0;


void 
read_p()
/*
 *	Read m, the degree of a primitive polynomial p(x) used to compute the
 *	Galois field GF(2**m). Get precomputed coefficients p[] of p(x). Read
 *	the code length.
 */
{
	int			i, ninf;

	printf("\nEnter a value of m such that the code length is\n");
	printf("2**(m-1) - 1 < length <= 2**m - 1\n\n");
    do {
	   printf("Enter m (between 2 and 15): ");
	   // printf("Enter m (between 2 and 20): ");
	   scanf("%d", &m);
    } while ( !(m>1) || !(m<16) );
    // } while ( !(m>1) || !(m<21) );
	for (i=1; i<m; i++)
		p[i] = 0;
	p[0] = p[m] = 1;
	if (m == 2)			p[1] = 1;
	else if (m == 3)	p[1] = 1;
	else if (m == 4)	p[1] = 1;
	else if (m == 5)	p[2] = 1;
	else if (m == 6)	p[1] = 1;
	else if (m == 7)	p[1] = 1;
	else if (m == 8)	p[4] = p[5] = p[6] = 1;
	else if (m == 9)	p[4] = 1;
	else if (m == 10)	p[3] = 1;
	else if (m == 11)	p[2] = 1;
	else if (m == 12)	p[3] = p[4] = p[7] = 1;
	else if (m == 13)	p[1] = p[3] = p[4] = 1;
	else if (m == 14)	p[1] = p[11] = p[12] = 1;
	else if (m == 15)	p[1] = 1;
	// else if (m == 16)	p[2] = p[3] = p[5] = 1;
	// else if (m == 17)	p[3] = 1;
	// else if (m == 18)	p[7] = 1;
	// else if (m == 19)	p[1] = p[5] = p[6] = 1;
	// else if (m == 20)	p[3] = 1;
	printf("p(x) = ");
    n = 1;
	for (i = 0; i <= m; i++) {
        n *= 2;
		printf("%1d", p[i]);
        }
	printf("\n");
	n = n / 2 - 1;
	ninf = (n + 1) / 2 - 1;
	do  {
		printf("Enter code length (%d < length <= %d): ", ninf, n);
		scanf("%d", &length);
	} while ( !((length <= n)&&(length>ninf)) );
}


void 
generate_gf()
/*
 * Generate field GF(2**m) from the irreducible polynomial p(X) with
 * coefficients in p[0]..p[m].
 *
 * Lookup tables:
 *   index->polynomial form: alpha_to[] contains j=alpha^i;
 *   polynomial form -> index form:	index_of[j=alpha^i] = i
 *
 * alpha=2 is the primitive element of GF(2**m) 
 */
{
	register int    i, mask;

	mask = 1;
	alpha_to[m] = 0;
	for (i = 0; i < m; i++) {
		alpha_to[i] = mask;
		index_of[alpha_to[i]] = i;
		if (p[i] != 0)
			alpha_to[m] ^= mask;
		mask <<= 1;
	}
	index_of[alpha_to[m]] = m;
	mask >>= 1;
	for (i = m + 1; i < n; i++) {
		if (alpha_to[i - 1] >= mask)
		  alpha_to[i] = alpha_to[m] ^ ((alpha_to[i - 1] ^ mask) << 1);
		else
		  alpha_to[i] = alpha_to[i - 1] << 1;
		index_of[alpha_to[i]] = i;
	}
	index_of[0] = -1;
}


void 
gen_poly()
/*
 * Compute the generator polynomial of a binary BCH code. Fist generate the
 * cycle sets modulo 2**m - 1, cycle[][] =  (i, 2*i, 4*i, ..., 2^l*i). Then
 * determine those cycle sets that contain integers in the set of (d-1)
 * consecutive integers {1..(d-1)}. The generator polynomial is calculated
 * as the product of linear factors of the form (x+alpha^i), for every i in
 * the above cycle sets.
 */
{
	register int	ii, jj, ll, kaux;
	register int	test, aux, nocycles, root, noterms, rdncy;
	int             cycle[1024][21], size[1024], min[1024], zeros[1024];

	/* Generate cycle sets modulo n, n = 2**m - 1 */
	cycle[0][0] = 0;
	size[0] = 1;
	cycle[1][0] = 1;
	size[1] = 1;
	jj = 1;			/* cycle set index */
	if (m > 9)  {
		printf("Computing cycle sets modulo %d\n", n);
		printf("(This may take some time)...\n");
	}
	do {
		/* Generate the jj-th cycle set */
		ii = 0;
		do {
			ii++;
			cycle[jj][ii] = (cycle[jj][ii - 1] * 2) % n;
			size[jj]++;
			aux = (cycle[jj][ii] * 2) % n;
		} while (aux != cycle[jj][0]);
		/* Next cycle set representative */
		ll = 0;
		do {
			ll++;
			test = 0;
			for (ii = 1; ((ii <= jj) && (!test)); ii++)	
			/* Examine previous cycle sets */
			  for (kaux = 0; ((kaux < size[ii]) && (!test)); kaux++)
			     if (ll == cycle[ii][kaux])
			        test = 1;
		} while ((test) && (ll < (n - 1)));
		if (!(test)) {
			jj++;	/* next cycle set index */
			cycle[jj][0] = ll;
			size[jj] = 1;
		}
	} while (ll < (n - 1));
	nocycles = jj;		/* number of cycle sets modulo n */

	printf("Enter the error correcting capability, t: ");
	scanf("%d", &t);

	d = 2 * t + 1;

	/* Search for roots 1, 2, ..., d-1 in cycle sets */
	kaux = 0;
	rdncy = 0;
	for (ii = 1; ii <= nocycles; ii++) {
		min[kaux] = 0;
		test = 0;
		for (jj = 0; ((jj < size[ii]) && (!test)); jj++)
			for (root = 1; ((root < d) && (!test)); root++)
				if (root == cycle[ii][jj])  {
					test = 1;
					min[kaux] = ii;
				}
		if (min[kaux]) {
			rdncy += size[min[kaux]];
			kaux++;
		}
	}
	noterms = kaux;
	kaux = 1;
	for (ii = 0; ii < noterms; ii++)
		for (jj = 0; jj < size[min[ii]]; jj++) {
			zeros[kaux] = cycle[min[ii]][jj];
			kaux++;
		}

	k = length - rdncy;

    if (k<0)
      {
         printf("Parameters invalid!\n");
         exit(0);
      }

	printf("This is a (%d, %d, %d) binary BCH code\n", length, k, d);

	/* Compute the generator polynomial */
	g[0] = alpha_to[zeros[1]];
	g[1] = 1;		/* g(x) = (X + zeros[1]) initially */
	for (ii = 2; ii <= rdncy; ii++) {
	  g[ii] = 1;
	  for (jj = ii - 1; jj > 0; jj--)
	    if (g[jj] != 0)
	      g[jj] = g[jj - 1] ^ alpha_to[(index_of[g[jj]] + zeros[ii]) % n];
	    else
	      g[jj] = g[jj - 1];
	  g[0] = alpha_to[(index_of[g[0]] + zeros[ii]) % n];
	}
	printf("Generator polynomial:\ng(x) = ");
	for (ii = 0; ii <= rdncy; ii++) {
	  printf("%d", g[ii]);
	  if (ii && ((ii % 50) == 0))
	    printf("\n");
	}
	printf("\n");
}


void 
encode_bch()
/*
 * Compute redundacy bb[], the coefficients of b(x). The redundancy
 * polynomial b(x) is the remainder after dividing x^(length-k)*data(x)
 * by the generator polynomial g(x).
 */
{
	register int    i, j;
	register int    feedback;

	for (i = 0; i < length - k; i++)
		bb[i] = 0;
	for (i = k - 1; i >= 0; i--) {
		feedback = data[i] ^ bb[length - k - 1];
		if (feedback != 0) {
			for (j = length - k - 1; j > 0; j--)
				if (g[j] != 0)
					bb[j] = bb[j - 1] ^ feedback;
				else
					bb[j] = bb[j - 1];
			bb[0] = g[0] && feedback;
		} else {
			for (j = length - k - 1; j > 0; j--)
				bb[j] = bb[j - 1];
			bb[0] = 0;
		}
	}
}


void 
decode_bch()
{
register int i, j, u, q, t2, count = 0, syn_error = 0;
int elp[1026][1024], l[1], s[1025];
int root[200], loc[200], err[1024], reg[201];
int qt[513], r[129][513];
int b[12][513];
int degr[129], degb[129];
int temp, aux[513];

  t2 = 2 * t;

  /* Compute the syndromes */
  printf("S(x) = ");
  for (i = 1; i <= t2; i++) {
    s[i] = 0;
    for (j = 0; j < length; j++)
      if (recd[j] != 0)
      s[i] ^= alpha_to[(i * j) % n];
      if (s[i] != 0)
      syn_error = 1; /* set error flag if non-zero syndrome */
      /* convert syndrome from polynomial form to index form  */
      s[i] = index_of[s[i]];
      printf("%3d ", s[i]);
    }