www.pudn.com > RS_Code.zip > rs.c, change:1996-09-27,size:15380b


/* 
 * 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]; 
 
/* 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) 
{ 
	generate_gf(); 
	gen_poly(); 
} 
 
/* 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]]; 
} 
 
 
/* 
 * 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 recd[NN]; 
	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; 
 
	/* data[] is in polynomial form, copy and convert to index form */ 
	for (i = NN-1; i >= 0; i--){ 
#if (MM != 8) 
		if(data[i] > NN) 
			return -1;	/* Illegal symbol */ 
#endif 
		recd[i] = Index_of[data[i]]; 
	} 
	/* first form the syndromes; i.e., evaluate recd(x) at roots of g(x) 
	 * namely @**(B0+i), i = 0, ... ,(NN-KK-1) 
	 */ 
	syn_error = 0; 
	for (i = 1; i <= NN-KK; i++) { 
		tmp = 0; 
		for (j = 0; j < NN; j++) 
			if (recd[j] != A0)	/* recd[j] in index form */ 
				tmp ^= Alpha_to[modnn(recd[j] + (B0+i-1)*j)]; 
		syn_error |= tmp;	/* set flag if non-zero syndrome => 
					 * error */ 
		/* store syndrome in index form  */ 
		s[i] = Index_of[tmp]; 
	} 
	if (!syn_error) { 
		/* 
		 * if syndrome is zero, data[] is a codeword and there are no 
		 * errors to correct. So return data[] unmodified 
		 */ 
		return 0; 
	} 
	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(®[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; 
}