Reed-Solomon coding and decoding by Phil Karn

/* * Reed-Solomon coding and decoding * Phil Karn (karn at ka9q.ampr.org) September 1996 * * This file is derived from the program "new_rs_erasures.c" by Robert * Morelos-Zaragoza (robert at spectra.eng.hawaii.edu) and Hari Thirumoorthy * (harit at 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, Error Control Coding Appendix A, * and Lee & Messerschmitt, Digital Communication 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(&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; }

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