rs.c

/*
 * Reed-Solomon coding and decoding
 * Phil Karn (karn@ka9q.ampr.org) September 1996
 * 
 * This file is derived from the program "new_rs_erasures.c" by Robert
 * Morelos-Zaragoza (robert@spectra.eng.hawaii.edu) and Hari Thirumoorthy
 * (harit@spectra.eng.hawaii.edu), Aug 1995
 *
 * I've made changes to improve performance, clean up the code and make it
 * easier to follow. Data is now passed to the encoding and decoding functions
 * through arguments rather than in global arrays. The decode function returns
 * the number of corrected symbols, or -1 if the word is uncorrectable.
 *
 * This code supports a symbol size from 2 bits up to 16 bits,
 * implying a block size of 3 2-bit symbols (6 bits) up to 65535
 * 16-bit symbols (1,048,560 bits). The code parameters are set in rs.h.
 *
 * Note that if symbols larger than 8 bits are used, the type of each
 * data array element switches from unsigned char to unsigned int. The
 * caller must ensure that elements larger than the symbol range are
 * not passed to the encoder or decoder.
 *
 */
#include <stdio.h>
#include "rs.h"

#if (KK >= NN)
#error "KK must be less than 2**MM - 1"
#endif

/* This defines the type used to store an element of the Galois Field
 * used by the code. Make sure this is something larger than a char if
 * if anything larger than GF(256) is used.
 *
 * Note: unsigned char will work up to GF(256) but int seems to run
 * faster on the Pentium.
 */
typedef int gf;

/* Primitive polynomials - see Lin & Costello, Appendix A,
 * and  Lee & Messerschmitt, p. 453.
 */
#if(MM == 2)/* Admittedly silly */
int Pp[MM+1] = { 1, 1, 1 };

#elif(MM == 3)
/* 1 + x + x^3 */
int Pp[MM+1] = { 1, 1, 0, 1 };

#elif(MM == 4)
/* 1 + x + x^4 */
int Pp[MM+1] = { 1, 1, 0, 0, 1 };

#elif(MM == 5)
/* 1 + x^2 + x^5 */
int Pp[MM+1] = { 1, 0, 1, 0, 0, 1 };

#elif(MM == 6)
/* 1 + x + x^6 */
int Pp[MM+1] = { 1, 1, 0, 0, 0, 0, 1 };

#elif(MM == 7)
/* 1 + x^3 + x^7 */
int Pp[MM+1] = { 1, 0, 0, 1, 0, 0, 0, 1 };

#elif(MM == 8)
/* 1+x^2+x^3+x^4+x^8 */
int Pp[MM+1] = { 1, 0, 1, 1, 1, 0, 0, 0, 1 };

#elif(MM == 9)
/* 1+x^4+x^9 */
int Pp[MM+1] = { 1, 0, 0, 0, 1, 0, 0, 0, 0, 1 };

#elif(MM == 10)
/* 1+x^3+x^10 */
int Pp[MM+1] = { 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1 };

#elif(MM == 11)
/* 1+x^2+x^11 */
int Pp[MM+1] = { 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1 };

#elif(MM == 12)
/* 1+x+x^4+x^6+x^12 */
int Pp[MM+1] = { 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1 };

#elif(MM == 13)
/* 1+x+x^3+x^4+x^13 */
int Pp[MM+1] = { 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1 };

#elif(MM == 14)
/* 1+x+x^6+x^10+x^14 */
int Pp[MM+1] = { 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1 };

#elif(MM == 15)
/* 1+x+x^15 */
int Pp[MM+1] = { 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 };

#elif(MM == 16)
/* 1+x+x^3+x^12+x^16 */
int Pp[MM+1] = { 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1 };

#else
#error "MM must be in range 2-16"
#endif

/* Alpha exponent for the first root of the generator polynomial */
#define B0	1

/* index->polynomial form conversion table */
gf Alpha_to[NN + 1];

/* Polynomial->index form conversion table */
gf Index_of[NN + 1];

/* No legal value in index form represents zero, so
 * we need a special value for this purpose
 */
#define A0	(NN)

/* Generator polynomial g(x)
 * Degree of g(x) = 2*TT
 * has roots @**B0, @**(B0+1), ... ,@^(B0+2*TT-1)
 */
gf Gg[NN - KK + 1];

/* Lookup tables for GF multiplication
 * Mtab[i][j] = j * alpha^i (note limited range of i)
 */
gf Mtab[NN-KK+1][NN+1];

/* Compute x % NN, where NN is 2**MM - 1,
 * without a slow divide
 */
static inline gf
modnn(int x)
{
	while (x >= NN) {
		x -= NN;
		x = (x >> MM) + (x & NN);
	}
	return x;
}

#define	min(a,b)	((a) < (b) ? (a) : (b))

#define	CLEAR(a,n) {\
	int ci;\
	for(ci=(n)-1;ci >=0;ci--)\
		(a)[ci] = 0;\
	}

#define	COPY(a,b,n) {\
	int ci;\
	for(ci=(n)-1;ci >=0;ci--)\
		(a)[ci] = (b)[ci];\
	}
#define	COPYDOWN(a,b,n) {\
	int ci;\
	for(ci=(n)-1;ci >=0;ci--)\
		(a)[ci] = (b)[ci];\
	}

void init_rs(void)
{
	void gen_mtab(void);
	generate_gf();
	gen_poly();
	gen_mtab();
}

/* generate GF(2**m) from the irreducible polynomial p(X) 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)
   HARI's COMMENT: (4/13/94) alpha_to[] can be used as follows:
        Let @ represent the primitive element commonly called "alpha" that
   is the root of the primitive polynomial p(x). Then in GF(2^m), for any
   0 <= i <= 2^m-2,
        @^i = a(0) + a(1) @ + a(2) @^2 + ... + a(m-1) @^(m-1)
   where the binary vector (a(0),a(1),a(2),...,a(m-1)) is the representation
   of the integer "alpha_to[i]" with a(0) being the LSB and a(m-1) the MSB. Thus for
   example the polynomial representation of @^5 would be given by the binary
   representation of the integer "alpha_to[5]".
                   Similarily, index_of[] can be used as follows:
        As above, let @ represent the primitive element of GF(2^m) that is
   the root of the primitive polynomial p(x). In order to find the power
   of @ (alpha) that has the polynomial representation
        a(0) + a(1) @ + a(2) @^2 + ... + a(m-1) @^(m-1)
   we consider the integer "i" whose binary representation with a(0) being LSB
   and a(m-1) MSB is (a(0),a(1),...,a(m-1)) and locate the entry
   "index_of[i]". Now, @^index_of[i] is that element whose polynomial 
    representation is (a(0),a(1),a(2),...,a(m-1)).
   NOTE:
        The element alpha_to[2^m-1] = 0 always signifying that the
   representation of "@^infinity" = 0 is (0,0,0,...,0).
        Similarily, the element index_of[0] = A0 always signifying
   that the power of alpha which has the polynomial representation
   (0,0,...,0) is "infinity".
 
*/

void
generate_gf(void)
{
	register int i, mask;

	mask = 1;
	Alpha_to[MM] = 0;
	for (i = 0; i < MM; i++) {
		Alpha_to[i] = mask;
		Index_of[Alpha_to[i]] = i;
		/* If Pp[i] == 1 then, term @^i occurs in poly-repr of @^MM */
		if (Pp[i] != 0)
			Alpha_to[MM] ^= mask;	/* Bit-wise EXOR operation */
		mask <<= 1;	/* single left-shift */
	}
	Index_of[Alpha_to[MM]] = MM;
	/*
	 * Have obtained poly-repr of @^MM. Poly-repr of @^(i+1) is given by
	 * poly-repr of @^i shifted left one-bit and accounting for any @^MM
	 * term that may occur when poly-repr of @^i is shifted.
	 */
	mask >>= 1;
	for (i = MM + 1; i < NN; i++) {
		if (Alpha_to[i - 1] >= mask)
			Alpha_to[i] = Alpha_to[MM] ^ ((Alpha_to[i - 1] ^ mask) << 1);
		else
			Alpha_to[i] = Alpha_to[i - 1] << 1;
		Index_of[Alpha_to[i]] = i;
	}
	Index_of[0] = A0;
	Alpha_to[NN] = 0;
}


/*
 * Obtain the generator polynomial of the TT-error correcting, length
 * NN=(2**MM -1) Reed Solomon code from the product of (X+@**(B0+i)), i = 0,
 * ... ,(2*TT-1)
 *
 * Examples:
 *
 * If B0 = 1, TT = 1. deg(g(x)) = 2*TT = 2.
 * g(x) = (x+@) (x+@**2)
 *
 * If B0 = 0, TT = 2. deg(g(x)) = 2*TT = 4.
 * g(x) = (x+1) (x+@) (x+@**2) (x+@**3)
 */
void
gen_poly(void)
{
	register int i, j;

	Gg[0] = Alpha_to[B0];
	Gg[1] = 1;		/* g(x) = (X+@**B0) initially */
	for (i = 2; i <= NN - KK; i++) {
		Gg[i] = 1;
		/*
		 * Below multiply (Gg[0]+Gg[1]*x + ... +Gg[i]x^i) by
		 * (@**(B0+i-1) + x)
		 */
		for (j = i - 1; j > 0; j--)
			if (Gg[j] != 0)
				Gg[j] = Gg[j - 1] ^ Alpha_to[modnn((Index_of[Gg[j]]) + B0 + i - 1)];
			else
				Gg[j] = Gg[j - 1];
		/* Gg[0] can never be zero */
		Gg[0] = Alpha_to[modnn((Index_of[Gg[0]]) + B0 + i - 1)];
	}
	/* convert Gg[] to index form for quicker encoding */
	for (i = 0; i <= NN - KK; i++)
		Gg[i] = Index_of[Gg[i]];
}

void
gen_mtab(void)
{
	int i,j;

	for(i=0;i<=NN-KK;i++){
		Mtab[i][0] = 0;
		for(j=1;j<=NN;j++){
			Mtab[i][j] = Alpha_to[modnn(Index_of[j] + i)];
		}
	}
}




/*
 * take the string of symbols in data[i], i=0..(k-1) and encode
 * systematically to produce NN-KK parity symbols in bb[0]..bb[NN-KK-1] data[]
 * is input and bb[] is output in polynomial form. Encoding is done by using
 * a feedback shift register with appropriate connections specified by the
 * elements of Gg[], which was generated above. Codeword is   c(X) =
 * data(X)*X**(NN-KK)+ b(X)
 */
int
encode_rs(dtype data[KK], dtype bb[NN-KK])
{
	register int i, j;
	gf feedback;

	CLEAR(bb,NN-KK);
	for (i = KK - 1; i >= 0; i--) {
#if (MM != 8)
		if(data[i] > NN)
			return -1;	/* Illegal symbol */
#endif
		feedback = Index_of[data[i] ^ bb[NN - KK - 1]];
		if (feedback != A0) {	/* feedback term is non-zero */
			for (j = NN - KK - 1; j > 0; j--)
				if (Gg[j] != A0)
					bb[j] = bb[j - 1] ^ Alpha_to[modnn(Gg[j] + feedback)];
				else
					bb[j] = bb[j - 1];
			bb[0] = Alpha_to[modnn(Gg[0] + feedback)];
		} else {	/* feedback term is zero. encoder becomes a
				 * single-byte shifter */
			for (j = NN - KK - 1; j > 0; j--)
				bb[j] = bb[j - 1];
			bb[0] = 0;
		}
	}
	return 0;
}


/*
 * Performs ERRORS+ERASURES decoding of RS codes. If decoding is successful,
 * writes the codeword into data[] itself. Otherwise data[] is unaltered.
 *
 * Return number of symbols corrected, or -1 if codeword is illegal
 * or uncorrectable.
 * 
 * First "no_eras" erasures are declared by the calling program. Then, the
 * maximum # of errors correctable is t_after_eras = floor((NN-KK-no_eras)/2).
 * If the number of channel errors is not greater than "t_after_eras" the
 * transmitted codeword will be recovered. Details of algorithm can be found
 * in R. Blahut's "Theory ... of Error-Correcting Codes".
 */
int
eras_dec_rs(dtype data[NN], int eras_pos[NN-KK], int no_eras)
{
	int deg_lambda, el, deg_omega;
	int i, j, r;
	gf u,q,tmp,num1,num2,den,discr_r;
	gf lambda[NN-KK + 1], s[NN-KK + 1];	/* Err+Eras Locator poly
						 * and syndrome poly */
	gf b[NN-KK + 1], t[NN-KK + 1], omega[NN-KK + 1];
	gf root[NN-KK], reg[NN-KK + 1], loc[NN-KK];
	int syn_error, count;

/*#define old */
#ifdef	old
	/* form the syndromes; i.e., evaluate data(x) at roots of g(x)
	 * namely @**(B0+i), i = 0, ... ,(NN-KK-1)
	 */
#if (MM != 8)
	if(data[0] > NN)
		return -1;
#endif
	for(i=1;i<=NN-KK;i++)
		s[i] = data[0];
	for(j=1;j<NN;j++){
#if (MM != 8)
		if(data[j] > NN)
			return -1;	/* Illegal symbol */
#endif
		if(data[j] == 0)
			continue;
		tmp = Index_of[data[j]];

		/* The following code is equivalent to
		 *
		 * for(i=1;i<=NN-KK;i++)
		 *	s[i] ^= Alpha_to[modnn(tmp + (B0+i-1)*j)];
		 *
		 * but is faster.
		 */
		tmp += B0*j;
		for(i=1;i<=NN-KK;i++){
			if(tmp >= NN)
				tmp -= NN;
			s[i] ^= Alpha_to[tmp];
			tmp += j;
		}
	}
#else
	for(i=1;i<=NN-KK;i++){
		tmp = data[NN-1];
		for(j=NN-2;j>=0;j--){
			tmp = data[j] ^ Mtab[i][tmp];
		}
		s[i] = tmp;
	}
#endif
	/* Convert syndromes to index form, checking for nonzero condition */
	syn_error = 0;
	for(i=1;i<=NN-KK;i++){
		syn_error |= s[i];
		s[i] = Index_of[s[i]];
	}

	if (!syn_error) {
		/*
		 * if syndrome is zero, data[] is a codeword and there are no
		 * errors to correct. So return data[] unmodified
		 */
		return 0;
	}
#if (NN-KK > 2)
	/* Check for and quickly correct the special case of a single error.
	 * This is indicated by
	 * s[i] / s[i+1] = @^-k for all i and some constant k,
	 *	 (s in polynomial form), or
	 * s[i] - s[i+1] = -k,
	 *	 (s in index form)
	 *
	 * See Clark & Cain, "Error Correction Coding for Digital
	 * Communications", p209-210.
	 */
	if(s[2] != A0 && s[1] != A0){
		tmp = modnn(s[2] - s[1] + NN);	/* s[] in index form */
		for(i=2;i<NN-KK;i++){
			if(s[i+1] == A0 || s[i] == A0
			 || modnn(s[i+1] - s[i] + NN) != tmp)
			break;
		}
		if(i == NN-KK){
			/* single error */
			data[tmp] ^= Alpha_to[modnn(s[1] - tmp + NN)];
			return 1;
		}
	}
#endif
	CLEAR(&lambda[1],NN-KK);
	lambda[0] = 1;
	if (no_eras > 0) {
		/* Init lambda to be the erasure locator polynomial */
		lambda[1] = Alpha_to[eras_pos[0]];
		for (i = 1; i < no_eras; i++) {
			u = eras_pos[i];
			for (j = i+1; j > 0; j--) {
				tmp = Index_of[lambda[j - 1]];
				if(tmp != A0)
					lambda[j] ^= Alpha_to[modnn(u + tmp)];
			}
		}
#ifdef ERASURE_DEBUG
		/* find roots of the erasure location polynomial */
		for(i=1;i<=no_eras;i++)
			reg[i] = Index_of[lambda[i]];
		count = 0;
		for (i = 1; i <= NN; i++) {
			q = 1;
			for (j = 1; j <= no_eras; j++)
				if (reg[j] != A0) {
					reg[j] = modnn(reg[j] + j);
					q ^= Alpha_to[reg[j]];
				}
			if (!q) {
				/* store root and error location
				 * number indices
				 */
				root[count] = i;
				loc[count] = NN - i;
				count++;
			}
		}
		if (count != no_eras) {
			printf("\n lambda(x) is WRONG\n");
			return -1;
		}
#ifndef NO_PRINT
		printf("\n Erasure positions as determined by roots of Eras Loc Poly:\n");
		for (i = 0; i < count; i++)
			printf("%d ", loc[i]);
		printf("\n");
#endif
#endif
	}
	for(i=0;i<NN-KK+1;i++)
		b[i] = Index_of[lambda[i]];

	/*
	 * Begin Berlekamp-Massey algorithm to determine error+erasure
	 * locator polynomial
	 */
	r = no_eras;
	el = no_eras;
	while (++r <= NN-KK) {	/* r is the step number */
		/* Compute discrepancy at the r-th step in poly-form */
		discr_r = 0;
		for (i = 0; i < r; i++){
			if ((lambda[i] != 0) && (s[r - i] != A0)) {
				discr_r ^= Alpha_to[modnn(Index_of[lambda[i]] + s[r - i])];
			}
		}
		discr_r = Index_of[discr_r];	/* Index form */
		if (discr_r == A0) {
			/* 2 lines below: B(x) <-- x*B(x) */
			COPYDOWN(&b[1],b,NN-KK);
			b[0] = A0;
		} else {
			/* 7 lines below: T(x) <-- lambda(x) - discr_r*x*b(x) */
			t[0] = lambda[0];
			for (i = 0 ; i < NN-KK; i++) {
				if(b[i] != A0)
					t[i+1] = lambda[i+1] ^ Alpha_to[modnn(discr_r + b[i])];
				else
					t[i+1] = lambda[i+1];
			}
			if (2 * el <= r + no_eras - 1) {
				el = r + no_eras - el;
				/*
				 * 2 lines below: B(x) <-- inv(discr_r) *
				 * lambda(x)
				 */
				for (i = 0; i <= NN-KK; i++)
					b[i] = (lambda[i] == 0) ? A0 : modnn(Index_of[lambda[i]] - discr_r + NN);
			} else {
				/* 2 lines below: B(x) <-- x*B(x) */
				COPYDOWN(&b[1],b,NN-KK);
				b[0] = A0;
			}
			COPY(lambda,t,NN-KK+1);
		}
	}

	/* Convert lambda to index form and compute deg(lambda(x)) */
	deg_lambda = 0;
	for(i=0;i<NN-KK+1;i++){
		lambda[i] = Index_of[lambda[i]];
		if(lambda[i] != A0)
			deg_lambda = i;
	}
	/*
	 * Find roots of the error+erasure locator polynomial. By Chien
	 * Search
	 */
	COPY(&reg[1],&lambda[1],NN-KK);
	count = 0;		/* Number of roots of lambda(x) */
	for (i = 1; i <= NN; i++) {
		q = 1;
		for (j = deg_lambda; j > 0; j--)
			if (reg[j] != A0) {
				reg[j] = modnn(reg[j] + j);
				q ^= Alpha_to[reg[j]];
			}
		if (!q) {
			/* store root (index-form) and error location number */
			root[count] = i;
			loc[count] = NN - i;
			count++;
		}
	}

#ifdef DEBUG
	printf("\n Final error positions:\t");
	for (i = 0; i < count; i++)
		printf("%d ", loc[i]);
	printf("\n");
#endif
	if (deg_lambda != count) {
		/*
		 * deg(lambda) unequal to number of roots => uncorrectable
		 * error detected
		 */
		return -1;
	}
	/*
	 * Compute err+eras evaluator poly omega(x) = s(x)*lambda(x) (modulo
	 * x**(NN-KK)). in index form. Also find deg(omega).
	 */
	deg_omega = 0;
	for (i = 0; i < NN-KK;i++){
		tmp = 0;
		j = (deg_lambda < i) ? deg_lambda : i;
		for(;j >= 0; j--){
			if ((s[i + 1 - j] != A0) && (lambda[j] != A0))
				tmp ^= Alpha_to[modnn(s[i + 1 - j] + lambda[j])];
		}
		if(tmp != 0)
			deg_omega = i;
		omega[i] = Index_of[tmp];
	}
	omega[NN-KK] = A0;

	/*
	 * Compute error values in poly-form. num1 = omega(inv(X(l))), num2 =
	 * inv(X(l))**(B0-1) and den = lambda_pr(inv(X(l))) all in poly-form
	 */
	for (j = count-1; j >=0; j--) {
		num1 = 0;
		for (i = deg_omega; i >= 0; i--) {
			if (omega[i] != A0)
				num1  ^= Alpha_to[modnn(omega[i] + i * root[j])];
		}
		num2 = Alpha_to[modnn(root[j] * (B0 - 1) + NN)];
		den = 0;

		/* lambda[i+1] for i even is the formal derivative lambda_pr of lambda[i] */
		for (i = min(deg_lambda,NN-KK-1) & ~1; i >= 0; i -=2) {
			if(lambda[i+1] != A0)
				den ^= Alpha_to[modnn(lambda[i+1] + i * root[j])];
		}
		if (den == 0) {
#ifdef DEBUG
			printf("\n ERROR: denominator = 0\n");
#endif
			return -1;
		}
		/* Apply error to data */
		if (num1 != 0) {
			data[loc[j]] ^= Alpha_to[modnn(Index_of[num1] + Index_of[num2] + NN - Index_of[den])];
		}
	}
	return count;
}