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+ /*
+ * The copyright in this software is being made available under the 2-clauses
+ * BSD License, included below. This software may be subject to other third
+ * party and contributor rights, including patent rights, and no such rights
+ * are granted under this license.
+ *
+ * Copyright (c) 2001-2003, David Janssens
+ * Copyright (c) 2002-2003, Yannick Verschueren
+ * Copyright (c) 2003-2005, Francois Devaux and Antonin Descampe
+ * Copyright (c) 2005, Herve Drolon, FreeImage Team
+ * Copyright (c) 2002-2005, Communications and remote sensing Laboratory, Universite catholique de Louvain, Belgium
+ * Copyright (c) 2005-2006, Dept. of Electronic and Information Engineering, Universita' degli Studi di Perugia, Italy
+ * All rights reserved.
+ *
+ * Redistribution and use in source and binary forms, with or without
+ * modification, are permitted provided that the following conditions
+ * are met:
+ * 1. Redistributions of source code must retain the above copyright
+ * notice, this list of conditions and the following disclaimer.
+ * 2. Redistributions in binary form must reproduce the above copyright
+ * notice, this list of conditions and the following disclaimer in the
+ * documentation and/or other materials provided with the distribution.
+ *
+ * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS `AS IS'
+ * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
+ * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
+ * ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
+ * LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
+ * CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
+ * SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
+ * INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
+ * CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
+ * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
+ * POSSIBILITY OF SUCH DAMAGE.
+ */
+
+#ifdef USE_JPWL
+
+/**
+@file rs.c
+@brief Functions used to compute the Reed-Solomon parity and check of byte arrays
+
+*/
+
+/**
+ * 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 <stdlib.h>
+#include "rs.h"
+
+/* 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;
+
+/* KK = number of information symbols */
+static int KK;
+
+/* 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 0 /* Different from the default 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];*/
+gf Gg[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(int k)
+{
+ KK = k;
+ if (KK >= NN) {
+ printf("KK must be less than 2**MM - 1\n");
+ exit(1);
+ }
+
+ 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]".
+ Similarly, 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).
+ Similarly, 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, dtype *bb)
+{
+ 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, int *eras_pos, 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];
+ /* Err+Eras Locator poly and syndrome poly */
+ /*gf lambda[NN-KK + 1], s[NN-KK + 1];
+ gf b[NN-KK + 1], t[NN-KK + 1], omega[NN-KK + 1];
+ gf root[NN-KK], reg[NN-KK + 1], loc[NN-KK];*/
+ gf lambda[NN + 1], s[NN + 1];
+ gf b[NN + 1], t[NN + 1], omega[NN + 1];
+ gf root[NN], reg[NN + 1], loc[NN];
+ 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;
+}
+
+
+#endif /* USE_JPWL */
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