forked from KolibriOS/kolibrios
595 lines
17 KiB
C
595 lines
17 KiB
C
/*
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* Copyright (c) 2001-2003, David Janssens
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* Copyright (c) 2002-2003, Yannick Verschueren
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* Copyright (c) 2003-2005, Francois Devaux and Antonin Descampe
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* Copyright (c) 2005, Hervé Drolon, FreeImage Team
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* Copyright (c) 2002-2005, Communications and remote sensing Laboratory, Universite catholique de Louvain, Belgium
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* Copyright (c) 2005-2006, Dept. of Electronic and Information Engineering, Universita' degli Studi di Perugia, Italy
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* All rights reserved.
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*
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* Redistribution and use in source and binary forms, with or without
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* modification, are permitted provided that the following conditions
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* are met:
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* 1. Redistributions of source code must retain the above copyright
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* notice, this list of conditions and the following disclaimer.
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* 2. Redistributions in binary form must reproduce the above copyright
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* notice, this list of conditions and the following disclaimer in the
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* documentation and/or other materials provided with the distribution.
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*
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* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS `AS IS'
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* AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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* ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
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* LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
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* CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
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* SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
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* INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
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* CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
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* ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
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* POSSIBILITY OF SUCH DAMAGE.
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*/
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#ifdef USE_JPWL
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/**
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@file rs.c
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@brief Functions used to compute the Reed-Solomon parity and check of byte arrays
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*/
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/**
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* Reed-Solomon coding and decoding
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* Phil Karn (karn@ka9q.ampr.org) September 1996
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*
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* This file is derived from the program "new_rs_erasures.c" by Robert
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* Morelos-Zaragoza (robert@spectra.eng.hawaii.edu) and Hari Thirumoorthy
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* (harit@spectra.eng.hawaii.edu), Aug 1995
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*
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* I've made changes to improve performance, clean up the code and make it
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* easier to follow. Data is now passed to the encoding and decoding functions
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* through arguments rather than in global arrays. The decode function returns
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* the number of corrected symbols, or -1 if the word is uncorrectable.
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*
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* This code supports a symbol size from 2 bits up to 16 bits,
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* implying a block size of 3 2-bit symbols (6 bits) up to 65535
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* 16-bit symbols (1,048,560 bits). The code parameters are set in rs.h.
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*
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* Note that if symbols larger than 8 bits are used, the type of each
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* data array element switches from unsigned char to unsigned int. The
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* caller must ensure that elements larger than the symbol range are
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* not passed to the encoder or decoder.
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*
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*/
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#include <stdio.h>
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#include <stdlib.h>
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#include "rs.h"
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/* This defines the type used to store an element of the Galois Field
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* used by the code. Make sure this is something larger than a char if
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* if anything larger than GF(256) is used.
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*
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* Note: unsigned char will work up to GF(256) but int seems to run
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* faster on the Pentium.
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*/
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typedef int gf;
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/* Primitive polynomials - see Lin & Costello, Appendix A,
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* and Lee & Messerschmitt, p. 453.
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*/
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#if(MM == 2)/* Admittedly silly */
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int Pp[MM+1] = { 1, 1, 1 };
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#elif(MM == 3)
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/* 1 + x + x^3 */
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int Pp[MM+1] = { 1, 1, 0, 1 };
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#elif(MM == 4)
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/* 1 + x + x^4 */
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int Pp[MM+1] = { 1, 1, 0, 0, 1 };
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#elif(MM == 5)
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/* 1 + x^2 + x^5 */
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int Pp[MM+1] = { 1, 0, 1, 0, 0, 1 };
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#elif(MM == 6)
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/* 1 + x + x^6 */
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int Pp[MM+1] = { 1, 1, 0, 0, 0, 0, 1 };
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#elif(MM == 7)
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/* 1 + x^3 + x^7 */
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int Pp[MM+1] = { 1, 0, 0, 1, 0, 0, 0, 1 };
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#elif(MM == 8)
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/* 1+x^2+x^3+x^4+x^8 */
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int Pp[MM+1] = { 1, 0, 1, 1, 1, 0, 0, 0, 1 };
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#elif(MM == 9)
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/* 1+x^4+x^9 */
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int Pp[MM+1] = { 1, 0, 0, 0, 1, 0, 0, 0, 0, 1 };
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#elif(MM == 10)
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/* 1+x^3+x^10 */
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int Pp[MM+1] = { 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1 };
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#elif(MM == 11)
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/* 1+x^2+x^11 */
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int Pp[MM+1] = { 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1 };
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#elif(MM == 12)
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/* 1+x+x^4+x^6+x^12 */
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int Pp[MM+1] = { 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1 };
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#elif(MM == 13)
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/* 1+x+x^3+x^4+x^13 */
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int Pp[MM+1] = { 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1 };
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#elif(MM == 14)
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/* 1+x+x^6+x^10+x^14 */
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int Pp[MM+1] = { 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1 };
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#elif(MM == 15)
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/* 1+x+x^15 */
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int Pp[MM+1] = { 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 };
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#elif(MM == 16)
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/* 1+x+x^3+x^12+x^16 */
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int Pp[MM+1] = { 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1 };
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#else
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#error "MM must be in range 2-16"
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#endif
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/* Alpha exponent for the first root of the generator polynomial */
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#define B0 0 /* Different from the default 1 */
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/* index->polynomial form conversion table */
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gf Alpha_to[NN + 1];
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/* Polynomial->index form conversion table */
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gf Index_of[NN + 1];
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/* No legal value in index form represents zero, so
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* we need a special value for this purpose
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*/
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#define A0 (NN)
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/* Generator polynomial g(x)
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* Degree of g(x) = 2*TT
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* has roots @**B0, @**(B0+1), ... ,@^(B0+2*TT-1)
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*/
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/*gf Gg[NN - KK + 1];*/
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gf Gg[NN - 1];
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/* Compute x % NN, where NN is 2**MM - 1,
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* without a slow divide
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*/
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static /*inline*/ gf
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modnn(int x)
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{
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while (x >= NN) {
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x -= NN;
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x = (x >> MM) + (x & NN);
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}
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return x;
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}
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/*#define min(a,b) ((a) < (b) ? (a) : (b))*/
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#define CLEAR(a,n) {\
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int ci;\
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for(ci=(n)-1;ci >=0;ci--)\
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(a)[ci] = 0;\
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}
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#define COPY(a,b,n) {\
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int ci;\
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for(ci=(n)-1;ci >=0;ci--)\
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(a)[ci] = (b)[ci];\
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}
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#define COPYDOWN(a,b,n) {\
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int ci;\
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for(ci=(n)-1;ci >=0;ci--)\
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(a)[ci] = (b)[ci];\
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}
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void init_rs(int k)
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{
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KK = k;
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if (KK >= NN) {
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printf("KK must be less than 2**MM - 1\n");
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exit(1);
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}
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generate_gf();
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gen_poly();
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}
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/* generate GF(2**m) from the irreducible polynomial p(X) in p[0]..p[m]
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lookup tables: index->polynomial form alpha_to[] contains j=alpha**i;
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polynomial form -> index form index_of[j=alpha**i] = i
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alpha=2 is the primitive element of GF(2**m)
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HARI's COMMENT: (4/13/94) alpha_to[] can be used as follows:
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Let @ represent the primitive element commonly called "alpha" that
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is the root of the primitive polynomial p(x). Then in GF(2^m), for any
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0 <= i <= 2^m-2,
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@^i = a(0) + a(1) @ + a(2) @^2 + ... + a(m-1) @^(m-1)
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where the binary vector (a(0),a(1),a(2),...,a(m-1)) is the representation
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of the integer "alpha_to[i]" with a(0) being the LSB and a(m-1) the MSB. Thus for
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example the polynomial representation of @^5 would be given by the binary
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representation of the integer "alpha_to[5]".
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Similarily, index_of[] can be used as follows:
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As above, let @ represent the primitive element of GF(2^m) that is
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the root of the primitive polynomial p(x). In order to find the power
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of @ (alpha) that has the polynomial representation
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a(0) + a(1) @ + a(2) @^2 + ... + a(m-1) @^(m-1)
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we consider the integer "i" whose binary representation with a(0) being LSB
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and a(m-1) MSB is (a(0),a(1),...,a(m-1)) and locate the entry
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"index_of[i]". Now, @^index_of[i] is that element whose polynomial
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representation is (a(0),a(1),a(2),...,a(m-1)).
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NOTE:
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The element alpha_to[2^m-1] = 0 always signifying that the
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representation of "@^infinity" = 0 is (0,0,0,...,0).
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Similarily, the element index_of[0] = A0 always signifying
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that the power of alpha which has the polynomial representation
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(0,0,...,0) is "infinity".
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*/
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void
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generate_gf(void)
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{
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register int i, mask;
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mask = 1;
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Alpha_to[MM] = 0;
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for (i = 0; i < MM; i++) {
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Alpha_to[i] = mask;
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Index_of[Alpha_to[i]] = i;
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/* If Pp[i] == 1 then, term @^i occurs in poly-repr of @^MM */
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if (Pp[i] != 0)
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Alpha_to[MM] ^= mask; /* Bit-wise EXOR operation */
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mask <<= 1; /* single left-shift */
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}
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Index_of[Alpha_to[MM]] = MM;
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/*
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* Have obtained poly-repr of @^MM. Poly-repr of @^(i+1) is given by
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* poly-repr of @^i shifted left one-bit and accounting for any @^MM
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* term that may occur when poly-repr of @^i is shifted.
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*/
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mask >>= 1;
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for (i = MM + 1; i < NN; i++) {
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if (Alpha_to[i - 1] >= mask)
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Alpha_to[i] = Alpha_to[MM] ^ ((Alpha_to[i - 1] ^ mask) << 1);
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else
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Alpha_to[i] = Alpha_to[i - 1] << 1;
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Index_of[Alpha_to[i]] = i;
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}
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Index_of[0] = A0;
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Alpha_to[NN] = 0;
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}
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/*
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* Obtain the generator polynomial of the TT-error correcting, length
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* NN=(2**MM -1) Reed Solomon code from the product of (X+@**(B0+i)), i = 0,
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* ... ,(2*TT-1)
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*
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* Examples:
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*
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* If B0 = 1, TT = 1. deg(g(x)) = 2*TT = 2.
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* g(x) = (x+@) (x+@**2)
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*
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* If B0 = 0, TT = 2. deg(g(x)) = 2*TT = 4.
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* g(x) = (x+1) (x+@) (x+@**2) (x+@**3)
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*/
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void
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gen_poly(void)
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{
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register int i, j;
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Gg[0] = Alpha_to[B0];
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Gg[1] = 1; /* g(x) = (X+@**B0) initially */
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for (i = 2; i <= NN - KK; i++) {
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Gg[i] = 1;
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/*
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* Below multiply (Gg[0]+Gg[1]*x + ... +Gg[i]x^i) by
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* (@**(B0+i-1) + x)
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*/
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for (j = i - 1; j > 0; j--)
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if (Gg[j] != 0)
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Gg[j] = Gg[j - 1] ^ Alpha_to[modnn((Index_of[Gg[j]]) + B0 + i - 1)];
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else
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Gg[j] = Gg[j - 1];
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/* Gg[0] can never be zero */
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Gg[0] = Alpha_to[modnn((Index_of[Gg[0]]) + B0 + i - 1)];
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}
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/* convert Gg[] to index form for quicker encoding */
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for (i = 0; i <= NN - KK; i++)
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Gg[i] = Index_of[Gg[i]];
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}
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/*
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* take the string of symbols in data[i], i=0..(k-1) and encode
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* systematically to produce NN-KK parity symbols in bb[0]..bb[NN-KK-1] data[]
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* is input and bb[] is output in polynomial form. Encoding is done by using
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* a feedback shift register with appropriate connections specified by the
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* elements of Gg[], which was generated above. Codeword is c(X) =
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* data(X)*X**(NN-KK)+ b(X)
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*/
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int
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encode_rs(dtype *data, dtype *bb)
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{
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register int i, j;
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gf feedback;
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CLEAR(bb,NN-KK);
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for (i = KK - 1; i >= 0; i--) {
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#if (MM != 8)
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if(data[i] > NN)
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return -1; /* Illegal symbol */
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#endif
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feedback = Index_of[data[i] ^ bb[NN - KK - 1]];
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if (feedback != A0) { /* feedback term is non-zero */
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for (j = NN - KK - 1; j > 0; j--)
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if (Gg[j] != A0)
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bb[j] = bb[j - 1] ^ Alpha_to[modnn(Gg[j] + feedback)];
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else
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bb[j] = bb[j - 1];
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bb[0] = Alpha_to[modnn(Gg[0] + feedback)];
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} else { /* feedback term is zero. encoder becomes a
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* single-byte shifter */
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for (j = NN - KK - 1; j > 0; j--)
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bb[j] = bb[j - 1];
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bb[0] = 0;
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}
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}
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return 0;
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}
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/*
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* Performs ERRORS+ERASURES decoding of RS codes. If decoding is successful,
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* writes the codeword into data[] itself. Otherwise data[] is unaltered.
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*
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* Return number of symbols corrected, or -1 if codeword is illegal
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* or uncorrectable.
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*
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* First "no_eras" erasures are declared by the calling program. Then, the
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* maximum # of errors correctable is t_after_eras = floor((NN-KK-no_eras)/2).
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* If the number of channel errors is not greater than "t_after_eras" the
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* transmitted codeword will be recovered. Details of algorithm can be found
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* in R. Blahut's "Theory ... of Error-Correcting Codes".
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*/
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int
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eras_dec_rs(dtype *data, int *eras_pos, int no_eras)
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{
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int deg_lambda, el, deg_omega;
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int i, j, r;
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gf u,q,tmp,num1,num2,den,discr_r;
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gf recd[NN];
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/* Err+Eras Locator poly and syndrome poly */
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/*gf lambda[NN-KK + 1], s[NN-KK + 1];
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gf b[NN-KK + 1], t[NN-KK + 1], omega[NN-KK + 1];
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gf root[NN-KK], reg[NN-KK + 1], loc[NN-KK];*/
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gf lambda[NN + 1], s[NN + 1];
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gf b[NN + 1], t[NN + 1], omega[NN + 1];
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gf root[NN], reg[NN + 1], loc[NN];
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int syn_error, count;
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/* data[] is in polynomial form, copy and convert to index form */
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for (i = NN-1; i >= 0; i--){
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#if (MM != 8)
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if(data[i] > NN)
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return -1; /* Illegal symbol */
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#endif
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recd[i] = Index_of[data[i]];
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}
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/* first form the syndromes; i.e., evaluate recd(x) at roots of g(x)
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* namely @**(B0+i), i = 0, ... ,(NN-KK-1)
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*/
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syn_error = 0;
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for (i = 1; i <= NN-KK; i++) {
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tmp = 0;
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for (j = 0; j < NN; j++)
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if (recd[j] != A0) /* recd[j] in index form */
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tmp ^= Alpha_to[modnn(recd[j] + (B0+i-1)*j)];
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syn_error |= tmp; /* set flag if non-zero syndrome =>
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* error */
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/* store syndrome in index form */
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s[i] = Index_of[tmp];
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}
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if (!syn_error) {
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/*
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* if syndrome is zero, data[] is a codeword and there are no
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* errors to correct. So return data[] unmodified
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*/
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return 0;
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}
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CLEAR(&lambda[1],NN-KK);
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lambda[0] = 1;
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if (no_eras > 0) {
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/* Init lambda to be the erasure locator polynomial */
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lambda[1] = Alpha_to[eras_pos[0]];
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for (i = 1; i < no_eras; i++) {
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u = eras_pos[i];
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for (j = i+1; j > 0; j--) {
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tmp = Index_of[lambda[j - 1]];
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if(tmp != A0)
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lambda[j] ^= Alpha_to[modnn(u + tmp)];
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}
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}
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#ifdef ERASURE_DEBUG
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/* find roots of the erasure location polynomial */
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for(i=1;i<=no_eras;i++)
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reg[i] = Index_of[lambda[i]];
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count = 0;
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for (i = 1; i <= NN; i++) {
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q = 1;
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for (j = 1; j <= no_eras; j++)
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if (reg[j] != A0) {
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reg[j] = modnn(reg[j] + j);
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q ^= Alpha_to[reg[j]];
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}
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if (!q) {
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/* store root and error location
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* number indices
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*/
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root[count] = i;
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loc[count] = NN - i;
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count++;
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}
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}
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if (count != no_eras) {
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printf("\n lambda(x) is WRONG\n");
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return -1;
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}
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#ifndef NO_PRINT
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printf("\n Erasure positions as determined by roots of Eras Loc Poly:\n");
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for (i = 0; i < count; i++)
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printf("%d ", loc[i]);
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printf("\n");
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#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;
|
|
}
|
|
|
|
|
|
#endif /* USE_JPWL */
|