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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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