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647 lines
23 KiB
C++
647 lines
23 KiB
C++
/*
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* Reed-Solomon -- Reed-Solomon encoder / decoder library
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*
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* Copyright (c) 2014 Hard Consulting Corporation.
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* Copyright (c) 2006 Phil Karn, KA9Q
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*
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* It may be used under the terms of the GNU Lesser General Public License (LGPL).
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*
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* Simplified version of https://github.com/pjkundert/ezpwd-reed-solomon which
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* seems to be the fastest open-source decoder.
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*
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*/
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#ifndef REEDSOLOMON_H
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#define REEDSOLOMON_H
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#include <algorithm>
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#include <array>
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#include <cstdint>
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#include <cstring>
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#include <type_traits>
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#include <vector>
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// Preprocessor defines available:
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//
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// EZPWD_NO_MOD_TAB -- define to force no "modnn" Galois modulo table acceleration
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//
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//#define EZPWD_NO_MOD_TAB
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namespace ReedSolomon {
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//
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// reed_solomon_base - Reed-Solomon codec generic base class
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//
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class reed_solomon_base {
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public:
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virtual size_t datum() const = 0; // a data element's bits
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virtual size_t symbol() const = 0; // a symbol's bits
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virtual int size() const = 0; // R-S block size (maximum total symbols)
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virtual int nroots() const = 0; // R-S roots (parity symbols)
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virtual int load() const = 0; // R-S net payload (data symbols)
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virtual ~reed_solomon_base() {}
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reed_solomon_base() {}
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//
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// {en,de}code -- Compute/Correct errors/erasures in a Reed-Solomon encoded container
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//
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/// For decode, optionally specify some known erasure positions (up to nroots()). If
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/// non-empty 'erasures' is provided, it contains the positions of each erasure. If a
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/// non-zero pointer to a 'position' vector is provided, its capacity will be increased to
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/// be capable of storing up to 'nroots()' ints; the actual deduced error locations will be
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/// returned.
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///
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/// RETURN VALUE
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///
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/// Return -1 on error. The encode returns the number of parity symbols produced;
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/// decode returns the number of symbols corrected. Both errors and erasures are included,
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/// so long as they are actually different than the deduced value. In other words, if a
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/// symbol is marked as an erasure but it actually turns out to be correct, it's index will
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/// NOT be included in the returned count, nor the modified erasure vector!
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///
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virtual int encode(const uint8_t *data, int len, uint8_t *parity) const = 0;
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virtual int decode1(uint8_t *data, int len, uint8_t *parity,
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const std::vector<int> &erasure = std::vector<int>(), std::vector<int> *position = 0) const = 0;
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int decode(uint8_t *data,
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int len,
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int pad = 0, // ignore 'pad' symbols at start of array
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const std::vector<int> &erasure = std::vector<int>(),
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std::vector<int> *position = 0) const
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{
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return decode1((uint8_t*)(data + pad), len, (uint8_t*)(data + len), erasure, position);
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}
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};
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//
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// gfpoly - default field polynomial generator functor.
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//
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template <int PLY>
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struct gfpoly {
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int operator()(int sr) const
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{
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if (sr == 0) {
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sr = 1;
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} else {
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sr <<= 1;
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if (sr & (1 << 8))
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sr ^= PLY;
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sr &= ((1 << 8) - 1);
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}
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return sr;
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}
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};
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//
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// class reed_solomon_tabs -- R-S tables common to all RS(NN,*) with same SYM, PRM and PLY
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//
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template <int PRM, class PLY>
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class reed_solomon_tabs : public reed_solomon_base {
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public:
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typedef uint8_t symbol_t;
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static const size_t DATUM = 8; // bits
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static const size_t SYMBOL = 8; // bits / symbol
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static const int MM = 8;
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static const int SIZE = (1 << 8) - 1; // maximum symbols in field
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static const int NN = SIZE;
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static const int A0 = SIZE;
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static const int MODS // modulo table: 1/2 the symbol size squared, up to 4k
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#if defined(EZPWD_NO_MOD_TAB)
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= 0;
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#else
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= 8 > 8 ? (1 << 12) : (1 << 8 << 8 / 2);
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#endif
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static int iprim; // initialized to -1, below
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protected:
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static std::array<uint8_t, NN + 1> alpha_to;
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static std::array<uint8_t, NN + 1> index_of;
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static std::array<uint8_t, MODS> mod_of;
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virtual ~reed_solomon_tabs() {}
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reed_solomon_tabs() : reed_solomon_base()
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{
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// Do init if not already done. We check one value which is initialized to -1; this is
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// safe, 'cause the value will not be set 'til the initializing thread has completely
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// initialized the structure. Worst case scenario: multiple threads will initialize
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// identically. No mutex necessary.
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if (iprim >= 0)
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return;
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// Generate Galois field lookup tables
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index_of[0] = A0; // log(zero) = -inf
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alpha_to[A0] = 0; // alpha**-inf = 0
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PLY poly;
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int sr = poly(0);
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for (int i = 0; i < NN; i++) {
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index_of[sr] = i;
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alpha_to[i] = sr;
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sr = poly(sr);
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}
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// If it's not primitive, raise exception or abort
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if (sr != alpha_to[0]) {
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abort();
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}
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// Generate modulo table for some commonly used (non-trivial) values
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for (int x = NN; x < NN + MODS; ++x)
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mod_of[x - NN] = _modnn(x);
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// Find prim-th root of 1, index form, used in decoding.
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int iptmp = 1;
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while (iptmp % PRM != 0)
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iptmp += NN;
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iprim = iptmp / PRM;
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}
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//
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// modnn -- modulo replacement for galois field arithmetics, optionally w/ table acceleration
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//
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// @x: the value to reduce (will never be -'ve)
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//
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// where
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// MM = number of bits per symbol
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// NN = (2^MM) - 1
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//
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// Simple arithmetic modulo would return a wrong result for values >= 3 * NN
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//
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uint8_t _modnn(int x) const
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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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uint8_t modnn(int x) const
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{
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while (x >= NN + MODS) {
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x -= NN;
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x = (x >> MM) + (x & NN);
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}
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if (MODS && x >= NN)
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x = mod_of[x - NN];
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return x;
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}
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};
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//
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// class reed_solomon - Reed-Solomon codec
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//
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// @TYP: A symbol datum; {en,de}code operates on arrays of these
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// @DATUM: Bits per datum (a TYP())
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// @SYM{BOL}, MM: Bits per symbol
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// @NN: Symbols per block (== (1<<MM)-1)
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// @alpha_to: log lookup table
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// @index_of: Antilog lookup table
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// @genpoly: Generator polynomial
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// @NROOTS: Number of generator roots = number of parity symbols
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// @FCR: First consecutive root, index form
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// @PRM: Primitive element, index form
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// @iprim: prim-th root of 1, index form
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// @PLY: The primitive generator polynominal functor
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//
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// All reed_solomon<T, ...> instances with the same template type parameters share a common
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// (static) set of alpha_to, index_of and genpoly tables. The first instance to be constructed
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// initializes the tables.
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//
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// Each specialized type of reed_solomon implements a specific encode/decode method
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// appropriate to its datum 'TYP'. When accessed via a generic reed_solomon_base pointer, only
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// access via "safe" (size specifying) containers or iterators is available.
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//
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template <int RTS, int FCR, int PRM, class PLY>
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class reed_solomon : public reed_solomon_tabs<PRM, PLY> {
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public:
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typedef reed_solomon_tabs<PRM, PLY> tabs_t;
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using tabs_t::A0;
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using tabs_t::DATUM;
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using tabs_t::MM;
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using tabs_t::NN;
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using tabs_t::SIZE;
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using tabs_t::SYMBOL;
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using tabs_t::iprim;
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using tabs_t::alpha_to;
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using tabs_t::index_of;
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using tabs_t::modnn;
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static const int NROOTS = RTS;
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static const int LOAD = SIZE - NROOTS; // maximum non-parity symbol payload
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protected:
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static std::array<uint8_t, NROOTS + 1> genpoly;
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public:
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virtual size_t datum() const { return DATUM; }
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virtual size_t symbol() const { return SYMBOL; }
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virtual int size() const { return SIZE; }
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virtual int nroots() const { return NROOTS; }
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virtual int load() const { return LOAD; }
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using reed_solomon_base::decode;
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virtual int decode1(uint8_t *data, int len, uint8_t *parity,
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const std::vector<int> &erasure = std::vector<int>(), std::vector<int> *position = 0) const
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{
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return decode_mask(data, len, parity, erasure, position);
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}
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//
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// decode_mask -- mask INP data into valid SYMBOL data
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//
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/// Incoming data may be in a variety of sizes, and may contain information beyond the
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/// R-S symbol capacity. For example, we might use a 6-bit R-S symbol to correct the lower
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/// 6 bits of an 8-bit data character. This would allow us to correct common substitution
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/// errors (such as '2' for '3', 'R' for 'T', 'n' for 'm').
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///
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int decode_mask(uint8_t *data, int len,
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uint8_t *parity = 0, // either 0, or pointer to all parity symbols
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const std::vector<int> &erasure = std::vector<int>(), std::vector<int> *position = 0) const
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{
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if (!parity) {
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len -= NROOTS;
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parity = data + len;
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}
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int corrects;
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if (!erasure.size() && !position) {
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// No erasures, and error position info not wanted.
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corrects = decode(data, len, parity);
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} else {
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// Either erasure location info specified, or resultant error position info wanted;
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// Prepare pos (a temporary, if no position vector provided), and copy any provided
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// erasure positions. After number of corrections is known, resize the position
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// vector. Thus, we use any supplied erasure info, and optionally return any
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// correction position info separately.
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std::vector<int> _pos;
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std::vector<int> &pos = position ? *position : _pos;
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pos.resize(std::max(size_t(NROOTS), erasure.size()));
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std::copy(erasure.begin(), erasure.end(), pos.begin());
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corrects = decode(data, len, parity, &pos.front(), erasure.size());
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if (corrects > int(pos.size())) {
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return -1;
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}
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pos.resize(std::max(0, corrects));
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}
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return corrects;
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}
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virtual ~reed_solomon()
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{
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}
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reed_solomon() : reed_solomon_tabs<PRM, PLY>()
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{
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// We check one element of the array; this is safe, 'cause the value will not be
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// initialized 'til the initializing thread has completely initialized the array. Worst
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// case scenario: multiple threads will initialize identically. No mutex necessary.
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if (genpoly[0])
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return;
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std::array<uint8_t, NROOTS + 1> tmppoly; // uninitialized
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// Form RS code generator polynomial from its roots. Only lower-index entries are
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// consulted, when computing subsequent entries; only index 0 needs initialization.
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tmppoly[0] = 1;
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for (int i = 0, root = FCR * PRM; i < NROOTS; i++, root += PRM) {
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tmppoly[i + 1] = 1;
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// Multiply tmppoly[] by @**(root + x)
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for (int j = i; j > 0; j--) {
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if (tmppoly[j] != 0)
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tmppoly[j] = tmppoly[j - 1] ^ alpha_to[modnn(index_of[tmppoly[j]] + root)];
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else
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tmppoly[j] = tmppoly[j - 1];
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}
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// tmppoly[0] can never be zero
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tmppoly[0] = alpha_to[modnn(index_of[tmppoly[0]] + root)];
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}
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// convert NROOTS entries of tmppoly[] to genpoly[] in index form for quicker encoding,
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// in reverse order so genpoly[0] is last element initialized.
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for (int i = NROOTS; i >= 0; --i)
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genpoly[i] = index_of[tmppoly[i]];
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}
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virtual int encode(const uint8_t *data, int len, uint8_t *parity) // at least nroots
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const
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{
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// Check length parameter for validity
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for (int i = 0; i < NROOTS; i++)
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parity[i] = 0;
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for (int i = 0; i < len; i++) {
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uint8_t feedback = index_of[data[i] ^ parity[0]];
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if (feedback != A0) {
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for (int j = 1; j < NROOTS; j++)
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parity[j] ^= alpha_to[modnn(feedback + genpoly[NROOTS - j])];
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}
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std::rotate(parity, parity + 1, parity + NROOTS);
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if (feedback != A0)
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parity[NROOTS - 1] = alpha_to[modnn(feedback + genpoly[0])];
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else
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parity[NROOTS - 1] = 0;
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}
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return NROOTS;
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}
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int decode(uint8_t *data, int len,
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uint8_t *parity, // Requires: at least NROOTS
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int *eras_pos = 0, // Capacity: at least NROOTS
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int no_eras = 0, // Maximum: at most NROOTS
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uint8_t *corr = 0) // Capacity: at least NROOTS
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const
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{
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typedef std::array<uint8_t, NROOTS> typ_nroots;
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typedef std::array<uint8_t, NROOTS + 1> typ_nroots_1;
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typedef std::array<int, NROOTS> int_nroots;
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typ_nroots_1 lambda{{0}};
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typ_nroots syn;
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typ_nroots_1 b;
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typ_nroots_1 t;
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typ_nroots_1 omega;
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int_nroots root;
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typ_nroots_1 reg;
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int_nroots loc;
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int count = 0;
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// Check length parameter and erasures for validity
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int pad = NN - NROOTS - len;
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if (no_eras) {
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if (no_eras > NROOTS) {
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return -1;
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}
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for (int i = 0; i < no_eras; ++i) {
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if (eras_pos[i] < 0 || eras_pos[i] >= len + NROOTS) {
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return -1;
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}
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}
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}
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// form the syndromes; i.e., evaluate data(x) at roots of g(x)
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for (int i = 0; i < NROOTS; i++)
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syn[i] = data[0];
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for (int j = 1; j < len; j++) {
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for (int i = 0; i < NROOTS; i++) {
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if (syn[i] == 0) {
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syn[i] = data[j];
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} else {
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syn[i] = data[j] ^ alpha_to[modnn(index_of[syn[i]] + (FCR + i) * PRM)];
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}
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}
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}
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for (int j = 0; j < NROOTS; j++) {
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for (int i = 0; i < NROOTS; i++) {
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if (syn[i] == 0) {
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syn[i] = parity[j];
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} else {
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syn[i] = parity[j] ^ alpha_to[modnn(index_of[syn[i]] + (FCR + i) * PRM)];
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}
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}
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}
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// Convert syndromes to index form, checking for nonzero condition
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uint8_t syn_error = 0;
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for (int i = 0; i < NROOTS; i++) {
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syn_error |= syn[i];
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syn[i] = index_of[syn[i]];
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}
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int deg_lambda = 0;
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int deg_omega = 0;
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int r = no_eras;
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int el = no_eras;
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if (!syn_error) {
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// if syndrome is zero, data[] is a codeword and there are no errors to correct.
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count = 0;
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goto finish;
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}
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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. Convert erasure positions
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// from index into data, to index into Reed-Solomon block.
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lambda[1] = alpha_to[modnn(PRM * (NN - 1 - (eras_pos[0] + pad)))];
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for (int i = 1; i < no_eras; i++) {
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uint8_t u = modnn(PRM * (NN - 1 - (eras_pos[i] + pad)));
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for (int j = i + 1; j > 0; j--) {
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uint8_t 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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}
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}
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for (int i = 0; i < NROOTS + 1; i++)
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b[i] = index_of[lambda[i]];
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//
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// Begin Berlekamp-Massey algorithm to determine error+erasure locator polynomial
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//
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while (++r <= NROOTS) { // r is the step number
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// Compute discrepancy at the r-th step in poly-form
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uint8_t discr_r = 0;
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for (int i = 0; i < r; i++) {
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if ((lambda[i] != 0) && (syn[r - i - 1] != A0)) {
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discr_r ^= alpha_to[modnn(index_of[lambda[i]] + syn[r - i - 1])];
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}
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}
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discr_r = index_of[discr_r]; // Index form
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if (discr_r == A0) {
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// 2 lines below: B(x) <-- x*B(x)
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// Rotate the last element of b[NROOTS+1] to b[0]
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std::rotate(b.begin(), b.begin() + NROOTS, b.end());
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b[0] = A0;
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} else {
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// 7 lines below: T(x) <-- lambda(x)-discr_r*x*b(x)
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t[0] = lambda[0];
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for (int i = 0; i < NROOTS; i++) {
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if (b[i] != A0) {
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t[i + 1] = lambda[i + 1] ^ alpha_to[modnn(discr_r + b[i])];
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} else
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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 (int i = 0; i <= NROOTS; i++) {
|
|
b[i] = ((lambda[i] == 0) ? A0 : modnn(index_of[lambda[i]] - discr_r + NN));
|
|
}
|
|
} else {
|
|
// 2 lines below: B(x) <-- x*B(x)
|
|
std::rotate(b.begin(), b.begin() + NROOTS, b.end());
|
|
b[0] = A0;
|
|
}
|
|
lambda = t;
|
|
}
|
|
}
|
|
|
|
// Convert lambda to index form and compute deg(lambda(x))
|
|
for (int i = 0; i < NROOTS + 1; i++) {
|
|
lambda[i] = index_of[lambda[i]];
|
|
if (lambda[i] != NN)
|
|
deg_lambda = i;
|
|
}
|
|
// Find roots of error+erasure locator polynomial by Chien search
|
|
reg = lambda;
|
|
count = 0; // Number of roots of lambda(x)
|
|
for (int i = 1, k = iprim - 1; i <= NN; i++, k = modnn(k + iprim)) {
|
|
uint8_t q = 1; // lambda[0] is always 0
|
|
for (int j = deg_lambda; j > 0; j--) {
|
|
if (reg[j] != A0) {
|
|
reg[j] = modnn(reg[j] + j);
|
|
q ^= alpha_to[reg[j]];
|
|
}
|
|
}
|
|
if (q != 0)
|
|
continue; // Not a root
|
|
// store root (index-form) and error location number
|
|
root[count] = i;
|
|
loc[count] = k;
|
|
// If we've already found max possible roots, abort the search to save time
|
|
if (++count == deg_lambda)
|
|
break;
|
|
}
|
|
if (deg_lambda != count) {
|
|
// deg(lambda) unequal to number of roots => uncorrectable error detected
|
|
count = -1;
|
|
goto finish;
|
|
}
|
|
//
|
|
// Compute err+eras evaluator poly omega(x) = s(x)*lambda(x) (modulo x**NROOTS). in
|
|
// index form. Also find deg(omega).
|
|
//
|
|
deg_omega = deg_lambda - 1;
|
|
for (int i = 0; i <= deg_omega; i++) {
|
|
uint8_t tmp = 0;
|
|
for (int j = i; j >= 0; j--) {
|
|
if ((syn[i - j] != A0) && (lambda[j] != A0))
|
|
tmp ^= alpha_to[modnn(syn[i - j] + lambda[j])];
|
|
}
|
|
omega[i] = index_of[tmp];
|
|
}
|
|
|
|
//
|
|
// Compute error values in poly-form. num1 = omega(inv(X(l))), num2 = inv(X(l))**(fcr-1)
|
|
// and den = lambda_pr(inv(X(l))) all in poly-form
|
|
//
|
|
for (int j = count - 1; j >= 0; j--) {
|
|
uint8_t num1 = 0;
|
|
for (int i = deg_omega; i >= 0; i--) {
|
|
if (omega[i] != A0)
|
|
num1 ^= alpha_to[modnn(omega[i] + i * root[j])];
|
|
}
|
|
uint8_t num2 = alpha_to[modnn(root[j] * (FCR - 1) + NN)];
|
|
uint8_t den = 0;
|
|
|
|
// lambda[i+1] for i even is the formal derivative lambda_pr of lambda[i]
|
|
for (int i = std::min(deg_lambda, NROOTS - 1) & ~1; i >= 0; i -= 2) {
|
|
if (lambda[i + 1] != A0) {
|
|
den ^= alpha_to[modnn(lambda[i + 1] + i * root[j])];
|
|
}
|
|
}
|
|
// Apply error to data. Padding ('pad' unused symbols) begin at index 0.
|
|
if (num1 != 0) {
|
|
if (loc[j] < pad) {
|
|
// If the computed error position is in the 'pad' (the unused portion of the
|
|
// R-S data capacity), then our solution has failed -- we've computed a
|
|
// correction location outside of the data and parity we've been provided!
|
|
count = -1;
|
|
goto finish;
|
|
}
|
|
|
|
uint8_t cor = alpha_to[modnn(index_of[num1] + index_of[num2] + NN - index_of[den])];
|
|
// Store the error correction pattern, if a correction buffer is available
|
|
if (corr)
|
|
corr[j] = cor;
|
|
// If a data/parity buffer is given and the error is inside the message or
|
|
// parity data, correct it
|
|
if (loc[j] < (NN - NROOTS)) {
|
|
if (data) {
|
|
data[loc[j] - pad] ^= cor;
|
|
}
|
|
} else if (loc[j] < NN) {
|
|
if (parity)
|
|
parity[loc[j] - (NN - NROOTS)] ^= cor;
|
|
}
|
|
}
|
|
}
|
|
|
|
finish:
|
|
if (eras_pos != NULL) {
|
|
for (int i = 0; i < count; i++)
|
|
eras_pos[i] = loc[i] - pad;
|
|
}
|
|
return count;
|
|
}
|
|
};
|
|
|
|
//
|
|
// Define the static reed_solomon...<...> members; allowed in header for template types.
|
|
//
|
|
// The reed_solomon_tags<...>::iprim < 0 is used to indicate to the first instance that the
|
|
// static tables require initialization.
|
|
//
|
|
template <int PRM, class PLY>
|
|
int reed_solomon_tabs<PRM, PLY>::iprim = -1;
|
|
|
|
template <int PRM, class PLY>
|
|
std::array<uint8_t, reed_solomon_tabs<PRM, PLY>::NN + 1> reed_solomon_tabs<PRM, PLY>::alpha_to;
|
|
|
|
template <int PRM, class PLY>
|
|
std::array<uint8_t, reed_solomon_tabs<PRM, PLY>::NN + 1> reed_solomon_tabs<PRM, PLY>::index_of;
|
|
template <int PRM, class PLY>
|
|
std::array<uint8_t, reed_solomon_tabs<PRM, PLY>::MODS> reed_solomon_tabs<PRM, PLY>::mod_of;
|
|
|
|
template <int RTS, int FCR, int PRM, class PLY>
|
|
std::array<uint8_t, reed_solomon<RTS, FCR, PRM, PLY>::NROOTS + 1> reed_solomon<RTS, FCR, PRM, PLY>::genpoly;
|
|
|
|
//
|
|
// RS( ... ) -- Define a reed-solomon codec
|
|
//
|
|
// @SYMBOLS: Total number of symbols; must be a power of 2 minus 1, eg 2^8-1 == 255
|
|
// @PAYLOAD: The maximum number of non-parity symbols, eg 253 ==> 2 parity symbols
|
|
// @POLY: A primitive polynomial appropriate to the SYMBOLS size
|
|
// @FCR: The first consecutive root of the Reed-Solomon generator polynomial
|
|
// @PRIM: The primitive root of the generator polynomial
|
|
//
|
|
|
|
//
|
|
// RS<SYMBOLS, PAYLOAD> -- Standard partial specializations for Reed-Solomon codec type access
|
|
//
|
|
// Normally, Reed-Solomon codecs are described with terms like RS(255,252). Obtain various
|
|
// standard Reed-Solomon codecs using macros of a similar form, eg. RS<255, 252>. Standard PLY,
|
|
// FCR and PRM values are provided for various SYMBOL sizes, along with appropriate basic types
|
|
// capable of holding all internal Reed-Solomon tabular data.
|
|
//
|
|
// In order to provide "default initialization" of const RS<...> types, a user-provided
|
|
// default constructor must be provided.
|
|
//
|
|
template <size_t SYMBOLS, size_t PAYLOAD>
|
|
struct RS;
|
|
template <size_t PAYLOAD>
|
|
struct RS<255, PAYLOAD> : public ReedSolomon::reed_solomon<(255) - (PAYLOAD), 0, 1, ReedSolomon::gfpoly<0x11d>>
|
|
{
|
|
RS()
|
|
: ReedSolomon::reed_solomon<(255) - (PAYLOAD), 0, 1, ReedSolomon::gfpoly<0x11d>>()
|
|
{
|
|
}
|
|
};
|
|
|
|
} // namespace ReedSolomon
|
|
|
|
#endif // REEDSOLOMON_H
|