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sdrangel/sdrbase/util/reedsolomon.h

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