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643 lines
25 KiB
C#
643 lines
25 KiB
C#
// /***************************************************************************
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// The Disc Image Chef
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// ----------------------------------------------------------------------------
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//
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// Filename : ReedSolomon.cs
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// Author(s) : Natalia Portillo <claunia@claunia.com>
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//
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// Component : Checksums.
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//
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// --[ Description ] ----------------------------------------------------------
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//
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// Calculates a Reed-Solomon.
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//
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// --[ License ] --------------------------------------------------------------
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//
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// This program is free software: you can redistribute it and/or modify
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// it under the terms of the GNU General Public License as
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// published by the Free Software Foundation, either version 3 of the
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// License, or (at your option) any later version.
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//
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// This program is distributed in the hope that it will be useful,
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// but WITHOUT ANY WARRANTY; without even the implied warranty of
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// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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// GNU General Public License for more details.
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//
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// You should have received a copy of the GNU General Public License
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// along with this program. If not, see <http://www.gnu.org/licenses/>.
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//
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// ----------------------------------------------------------------------------
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// Copyright © 2011-2017 Natalia Portillo
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// Copyright (C) 1996 Phil Karn
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// Copyright (C) 1995 Robert Morelos-Zaragoza
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// Copyright (C) 1995 Hari Thirumoorthy
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// ****************************************************************************/
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/*
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* Reed-Solomon coding and decoding
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* Phil Karn (karn at 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 at spectra.eng.hawaii.edu) and Hari Thirumoorthy
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* (harit at 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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using System;
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using DiscImageChef.Console;
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namespace DiscImageChef.Checksums
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{
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public class ReedSolomon
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{
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/* Primitive polynomials - see Lin & Costello, Error Control Coding Appendix A,
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* and Lee & Messerschmitt, Digital Communication p. 453.
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*/
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int[] Pp;
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/* index->polynomial form conversion table */
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int[] Alpha_to;
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/* Polynomial->index form conversion table */
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int[] Index_of;
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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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int[] Gg;
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int MM, KK, NN;
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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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int A0;
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bool initialized;
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/* Alpha exponent for the first root of the generator polynomial */
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const int B0 = 1;
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/// <summary>
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/// Initializes the Reed-Solomon with RS(n,k) with GF(2^m)
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/// </summary>
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public void InitRS(int n, int k, int m)
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{
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switch(m)
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{
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case 2:
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Pp = new[] { 1, 1, 1 };
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break;
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case 3:
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Pp = new[] { 1, 1, 0, 1 };
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break;
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case 4:
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Pp = new[] { 1, 1, 0, 0, 1 };
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break;
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case 5:
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Pp = new[] { 1, 0, 1, 0, 0, 1 };
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break;
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case 6:
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Pp = new[] { 1, 1, 0, 0, 0, 0, 1 };
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break;
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case 7:
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Pp = new[] { 1, 0, 0, 1, 0, 0, 0, 1 };
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break;
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case 8:
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Pp = new[] { 1, 0, 1, 1, 1, 0, 0, 0, 1 };
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break;
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case 9:
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Pp = new[] { 1, 0, 0, 0, 1, 0, 0, 0, 0, 1 };
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break;
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case 10:
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Pp = new[] { 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1 };
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break;
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case 11:
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Pp = new[] { 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1 };
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break;
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case 12:
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Pp = new[] { 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1 };
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break;
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case 13:
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Pp = new[] { 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1 };
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break;
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case 14:
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Pp = new[] { 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1 };
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break;
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case 15:
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Pp = new[] { 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 };
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break;
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case 16:
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Pp = new[] { 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1 };
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break;
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default:
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throw new ArgumentOutOfRangeException(nameof(m), "m must be between 2 and 16 inclusive");
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}
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MM = m;
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KK = k;
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NN = n;
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A0 = n;
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Alpha_to = new int[n + 1];
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Index_of = new int[n + 1];
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Gg = new int[NN - KK + 1];
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generate_gf();
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gen_poly();
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initialized = true;
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}
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int modnn(int x)
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{
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while(x >= NN)
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{
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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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static int min(int a, int b)
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{
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return ((a) < (b) ? (a) : (b));
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}
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static void CLEAR(ref int[] a, int n)
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{
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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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static void COPY(ref int[] a, ref int[] b, int n)
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{
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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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static void COPYDOWN(ref int[] a, ref int[] b, int n)
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{
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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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/* 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 generate_gf()
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{
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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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{
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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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{
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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 gen_poly()
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{
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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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{
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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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/// <summary>
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/// Takes the symbols in data to output parity in bb.
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/// </summary>
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/// <returns>Returns -1 if an illegal symbol is found.</returns>
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/// <param name="data">Data symbols.</param>
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/// <param name="bb">Outs parity symbols.</param>
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public int encode_rs(int[] data, out int[] bb)
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{
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if(initialized)
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{
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int i, j;
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int feedback;
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bb = new int[NN - KK];
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CLEAR(ref bb, NN - KK);
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for(i = KK - 1; i >= 0; i--)
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{
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if(MM != 8)
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{
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if(data[i] > NN)
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return -1; /* Illegal symbol */
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}
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feedback = Index_of[data[i] ^ bb[NN - KK - 1]];
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if(feedback != A0)
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{ /* 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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}
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else
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{ /* 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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throw new UnauthorizedAccessException("Trying to calculate RS without initializing!");
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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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/// <summary>
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/// Decodes the RS. If decoding is successful outputs corrected data symbols.
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/// </summary>
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/// <returns>Returns corrected symbols, -1 if illegal or uncorrectable</returns>
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/// <param name="data">Data symbols.</param>
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/// <param name="eras_pos">Position of erasures.</param>
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/// <param name="no_eras">Number of erasures.</param>
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public int eras_dec_rs(ref int[] data, out int[] eras_pos, int no_eras)
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{
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if(initialized)
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{
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eras_pos = new int[NN - KK];
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int deg_lambda, el, deg_omega;
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int i, j, r;
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int u, q, tmp, num1, num2, den, discr_r;
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int[] recd = new int[NN];
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int[] lambda = new int[NN - KK + 1]; /* Err+Eras Locator poly */
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int[] s = new int[NN - KK + 1]; /* syndrome poly */
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int[] b = new int[NN - KK + 1];
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int[] t = new int[NN - KK + 1];
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int[] omega = new int[NN - KK + 1];
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int[] root = new int[NN - KK];
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int[] reg = new int[NN - KK + 1];
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int[] loc = new int[NN - KK];
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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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{
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if(MM != 8)
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{
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if(data[i] > NN)
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return -1; /* Illegal symbol */
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}
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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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{
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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 == 0)
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{
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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(ref lambda, NN - KK);
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lambda[0] = 1;
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if(no_eras > 0)
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{
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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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{
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u = eras_pos[i];
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for(j = i + 1; j > 0; j--)
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{
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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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#if 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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{
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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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{
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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 == 0)
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{
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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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{
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DicConsole.DebugWriteLine("Reed Solomon", "\n lambda(x) is WRONG\n");
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return -1;
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}
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DicConsole.DebugWriteLine("Reed Solomon", "\n Erasure positions as determined by roots of Eras Loc Poly:\n");
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for(i = 0; i < count; i++)
|
|
DicConsole.DebugWriteLine("Reed Solomon", "{0} ", loc[i]);
|
|
DicConsole.DebugWriteLine("Reed Solomon", "\n");
|
|
#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(ref b, ref 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(ref b, ref b, NN - KK);
|
|
b[0] = A0;
|
|
}
|
|
COPY(ref lambda, ref 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
|
|
*/
|
|
int temp = reg[0];
|
|
COPY(ref reg, ref lambda, NN - KK);
|
|
reg[0] = temp;
|
|
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 == 0)
|
|
{
|
|
/* store root (index-form) and error location number */
|
|
root[count] = i;
|
|
loc[count] = NN - i;
|
|
count++;
|
|
}
|
|
}
|
|
|
|
#if DEBUG
|
|
DicConsole.DebugWriteLine("Reed Solomon", "\n Final error positions:\t");
|
|
for(i = 0; i < count; i++)
|
|
DicConsole.DebugWriteLine("Reed Solomon", "{0} ", loc[i]);
|
|
DicConsole.DebugWriteLine("Reed Solomon", "\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)
|
|
{
|
|
DicConsole.DebugWriteLine("Reed Solomon", "\n ERROR: denominator = 0\n");
|
|
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;
|
|
}
|
|
throw new UnauthorizedAccessException("Trying to calculate RS without initializing!");
|
|
}
|
|
}
|
|
}
|