reedsolomon.cpp
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#include <itpp/comm/reedsolomon.h>
#include <itpp/base/specmat.h>
#include <itpp/base/math/log_exp.h>
namespace itpp
{
//-------------------- Help Function ----------------------------
GFX formal_derivate(const GFX &f)
{
int degree = f.get_true_degree();
int q = f.get_size();
int i;
GFX fprim(q, degree);
fprim.clear();
for (i = 0; i <= degree - 1; i += 2) {
fprim[i] = f[i + 1];
}
return fprim;
}
//-------------------- Reed-Solomon ----------------------------
//A Reed-Solomon code is a q^m-ary BCH code of length n = pow(q,m)-1.
//k = pow(q,m)-1-t. This class works for q==2.
Reed_Solomon::Reed_Solomon(int in_m, int in_t, bool sys, int in_b):
m(in_m), t(in_t), b(in_b), systematic(sys)
{
n = pow2i(m) - 1;
k = pow2i(m) - 1 - 2 * t;
q = pow2i(m);
it_assert( (b >= 0) && (b < n), "Reed_Solomon::Reed_Solomon: narrow-sense parameter restricted to 0 <= b <= n.");
GFX x(q, (char *)"-1 0");
ivec alphapow(1);
g.set(q, (char *)"0");
for (int i = 1; i <= 2 * t; i++) {
alphapow(0) = b + i - 1;
g *= (x - GFX(q, alphapow));
}
}
void Reed_Solomon::encode(const bvec &uncoded_bits, bvec &coded_bits)
{
int i, j, iterations = floor_i(static_cast<double>(uncoded_bits.length())
/ (k * m));
GFX mx(q, k), cx(q, n);
GFX r(n + 1, n - k);
GFX uncoded_shifted(n + 1, n);
GF mpow;
bvec mbit(k * m), cbit(m);
coded_bits.set_size(iterations * n * m, false);
if (systematic)
for (i = 0; i < n - k; i++)
uncoded_shifted[i] = GF(n + 1, -1);
for (i = 0; i < iterations; i++) {
//Fix the message polynom m(x).
for (j = 0; j < k; j++) {
mpow.set(q, uncoded_bits.mid((i * m * k) + (j * m), m));
mx[j] = mpow;
if (systematic) {
cx[j] = mx[j];
uncoded_shifted[j + n - k] = mx[j];
}
}
//Fix the outputbits cbit.
if (systematic) {
r = modgfx(uncoded_shifted, g);
for (j = k; j < n; j++) {
cx[j] = r[j - k];
}
}
else {
cx = g * mx;
}
for (j = 0; j < n; j++) {
cbit = cx[j].get_vectorspace();
coded_bits.replace_mid((i * n * m) + (j * m), cbit);
}
}
}
bvec Reed_Solomon::encode(const bvec &uncoded_bits)
{
bvec coded_bits;
encode(uncoded_bits, coded_bits);
return coded_bits;
}
bool Reed_Solomon::decode(const bvec &coded_bits, const ivec &erasure_positions, bvec &decoded_message, bvec &cw_isvalid)
{
bool decoderfailure, no_dec_failure;
int j, i, kk, l, L, foundzeros, iterations = floor_i(static_cast<double>(coded_bits.length()) / (n * m));
bvec mbit(m * k);
decoded_message.set_size(iterations * k * m, false);
cw_isvalid.set_length(iterations);
GFX rx(q, n - 1), cx(q, n - 1), mx(q, k - 1), ex(q, n - 1), S(q, 2 * t), Xi(q, 2 * t), Gamma(q), Lambda(q),
Psiprime(q), OldLambda(q), T(q), Omega(q);
GFX dummy(q), One(q, (char*)"0"), Omegatemp(q);
GF delta(q), tempsum(q), rtemp(q), temp(q), Xk(q), Xkinv(q);
ivec errorpos;
if ( erasure_positions.length() ) {
it_assert(max(erasure_positions) < iterations*n, "Reed_Solomon::decode: erasure position is invalid.");
}
no_dec_failure = true;
for (i = 0; i < iterations; i++) {
decoderfailure = false;
//Fix the received polynomial r(x)
for (j = 0; j < n; j++) {
rtemp.set(q, coded_bits.mid(i * n * m + j * m, m));
rx[j] = rtemp;
}
// Fix the Erasure polynomial Gamma(x)
// and replace erased coordinates with zeros
rtemp.set(q, -1);
ivec alphapow = - ones_i(2);
Gamma = One;
for (j = 0; j < erasure_positions.length(); j++) {
rx[erasure_positions(j)] = rtemp;
alphapow(1) = erasure_positions(j);
Gamma *= (One - GFX(q, alphapow));
}
//Fix the syndrome polynomial S(x).
S.clear();
for (j = 1; j <= 2 * t; j++) {
S[j] = rx(GF(q, b + j - 1));
}
// calculate the modified syndrome polynomial Xi(x) = Gamma * (1+S) - 1
Xi = Gamma * (One + S) - One;
// Apply Berlekam-Massey algorithm
if (Xi.get_true_degree() >= 1) { //Errors in the received word
// Iterate to find Lambda(x), which hold all error locations
kk = 0;
Lambda = One;
L = 0;
T = GFX(q, (char*)"-1 0");
while (kk < 2 * t) {
kk = kk + 1;
tempsum = GF(q, -1);
for (l = 1; l <= L; l++) {
tempsum += Lambda[l] * Xi[kk - l];
}
delta = Xi[kk] - tempsum;
if (delta != GF(q, -1)) {
OldLambda = Lambda;
Lambda -= delta * T;
if (2 * L < kk) {
L = kk - L;
T = OldLambda / delta;
}
}
T = GFX(q, (char*)"-1 0") * T;
}
// Find the zeros to Lambda(x)
errorpos.set_size(Lambda.get_true_degree());
foundzeros = 0;
for (j = q - 2; j >= 0; j--) {
if (Lambda(GF(q, j)) == GF(q, -1)) {
errorpos(foundzeros) = (n - j) % n;
foundzeros += 1;
if (foundzeros >= Lambda.get_true_degree()) {
break;
}
}
}
if (foundzeros != Lambda.get_true_degree()) {
decoderfailure = true;
}
else { // Forney algorithm...
//Compute Omega(x) using the key equation for RS-decoding
Omega.set_degree(2 * t);
Omegatemp = Lambda * (One + Xi);
for (j = 0; j <= 2 * t; j++) {
Omega[j] = Omegatemp[j];
}
//Find the error/erasure magnitude polynomial by treating them the same
Psiprime = formal_derivate(Lambda*Gamma);
errorpos = concat(errorpos, erasure_positions);
ex.clear();
for (j = 0; j < errorpos.length(); j++) {
Xk = GF(q, errorpos(j));
Xkinv = GF(q, 0) / Xk;
// we calculate ex = - error polynomial, in order to avoid the
// subtraction when recunstructing the corrected codeword
ex[errorpos(j)] = (Xk * Omega(Xkinv)) / Psiprime(Xkinv);
if (b != 1) { // non-narrow-sense code needs corrected error magnitudes
int correction_exp = ( errorpos(j)*(1-b) ) % n;
ex[errorpos(j)] *= GF(q, correction_exp + ( (correction_exp < 0) ? n : 0 ));
}
}
//Reconstruct the corrected codeword.
// instead of subtracting the error/erasures, we calculated
// the negative error with 'ex' above
cx = rx + ex;
//Code word validation
S.clear();
for (j = 1; j <= 2 * t; j++) {
S[j] = cx(GF(q, b + j - 1));
}
if (S.get_true_degree() >= 1) {
decoderfailure = true;
}
}
}
else {
cx = rx;
decoderfailure = false;
}
//Find the message polynomial
mbit.clear();
if (decoderfailure == false) {
if (cx.get_true_degree() >= 1) { // A nonzero codeword was transmitted
if (systematic) {
for (j = 0; j < k; j++) {
mx[j] = cx[j];
}
}
else {
mx = divgfx(cx, g);
}
for (j = 0; j <= mx.get_true_degree(); j++) {
mbit.replace_mid(j * m, mx[j].get_vectorspace());
}
}
}
else { //Decoder failure.
// for a systematic code it is better to extract the undecoded message
// from the received code word, i.e. obtaining a bit error
// prob. p_b << 1/2, than setting all-zero (p_b = 1/2)
if (systematic) {
mbit = coded_bits.mid(i * n * m, k * m);
}
else {
mbit = zeros_b(k);
}
no_dec_failure = false;
}
decoded_message.replace_mid(i * m * k, mbit);
cw_isvalid(i) = (!decoderfailure);
}
return no_dec_failure;
}
bool Reed_Solomon::decode(const bvec &coded_bits, bvec &decoded_message, bvec &cw_isvalid)
{
ivec erasures(0);
return decode(coded_bits, erasures, decoded_message, cw_isvalid);
}
void Reed_Solomon::decode(const bvec &coded_bits, bvec &decoded_bits)
{
bvec cw_isvalid;
ivec erasures(0);
if (!decode(coded_bits, erasures, decoded_bits, cw_isvalid)) {
for (int i = 0; i < cw_isvalid.length(); i++) {
if (!cw_isvalid(i)) {
decoded_bits.replace_mid(i * k * m, zeros_b(k * m));
}
}
}
}
bvec Reed_Solomon::decode(const bvec &coded_bits)
{
bvec decoded_bits;
decode(coded_bits, decoded_bits);
return decoded_bits;
}
// --- Soft-decision decoding is not implemented ---
void Reed_Solomon::decode(const vec &, bvec &)
{
it_error("Reed_Solomon::decode(): Soft-decision decoding not implemented");
}
bvec Reed_Solomon::decode(const vec &)
{
it_error("Reed_Solomon::decode(): Soft-decision decoding not implemented");
return bvec();
}
} // namespace itpp
itpp::GFX
Polynomials over GF(q)[x], where q=2^m, m=1,...,16.
Definition galois.h:176
itpp::GFX::get_true_degree
int get_true_degree() const
Return true degree of GF(q)[x].
Definition galois.h:466
itpp::GFX::set_degree
void set_degree(int indegree, bool copy=false)
Resize the polynomial to the given indegree. If copy is set to true, the old polynomial's coefficient...
Definition galois.h:459
itpp::GFX::get_size
int get_size() const
Return q.
Definition galois.h:449
itpp::GFX::clear
void clear()
Set all coefficients to zero.
Definition galois.h:477
itpp::GF
Galois Field GF(q).
Definition galois.h:75
itpp::GF::set
void set(int qvalue, int inexp)
GF(q) equals alpha ^ inexp.
Definition galois.h:92
itpp::Reed_Solomon::t
int t
Definition reedsolomon.h:115
itpp::Reed_Solomon::encode
virtual void encode(const bvec &uncoded_bits, bvec &coded_bits)
Encoder function.
Definition reedsolomon.cpp:71
itpp::Reed_Solomon::n
int n
Definition reedsolomon.h:115
itpp::Reed_Solomon::decode
virtual bool decode(const bvec &coded_bits, const ivec &erasure_positions, bvec &decoded_message, bvec &cw_isvalid)
Decode the RS code bits. Return false if there has been any decoding failure.
Definition reedsolomon.cpp:121
itpp::Reed_Solomon::b
int b
Definition reedsolomon.h:115
itpp::Reed_Solomon::Reed_Solomon
Reed_Solomon(int in_m, int in_t, bool sys=false, int in_b=1)
Definition reedsolomon.cpp:55
itpp::Reed_Solomon::q
int q
Definition reedsolomon.h:115
itpp::Reed_Solomon::g
GFX g
The generator polynomial of the RS code.
Definition reedsolomon.h:118
itpp::Reed_Solomon::k
int k
Definition reedsolomon.h:115
itpp::Reed_Solomon::m
int m
Definition reedsolomon.h:115
itpp::Reed_Solomon::systematic
const bool systematic
Whether or not the code is systematic.
Definition reedsolomon.h:120
it_error
#define it_error(s)
Abort unconditionally.
Definition itassert.h:126
it_assert
#define it_assert(t, s)
Abort if t is not true.
Definition itassert.h:94
itpp::pow2i
int pow2i(int x)
Calculate two to the power of x (2^x); x is integer.
Definition log_exp.h:53
itpp::max
T max(const Vec< T > &v)
Maximum value of vector.
Definition min_max.h:45
itpp::ones_i
ITPP_EXPORT ivec ones_i(int size)
A Int vector of ones.
itpp::zeros_b
ITPP_EXPORT bvec zeros_b(int size)
A Binary vector of zeros.
log_exp.h
Logarithmic and exponenential functions - header file.
itpp
itpp namespace
Definition itmex.h:37
itpp::concat
const Array< T > concat(const Array< T > &a, const T &e)
Append element e to the end of the Array a.
Definition array.h:486
itpp::formal_derivate
GFX formal_derivate(const GFX &f)
Local help function.
Definition reedsolomon.cpp:39
itpp::divgfx
GFX divgfx(const GFX &c, const GFX &g)
Division of two GFX (local help function).
Definition galois.cpp:157
itpp::floor_i
int floor_i(double x)
The nearest smaller integer.
Definition converters.h:350
itpp::modgfx
GFX modgfx(const GFX &a, const GFX &b)
Modulo function of two GFX (local help function).
Definition galois.cpp:183
reedsolomon.h
Definitions of a Reed-Solomon codec class.
specmat.h
Definitions of special vectors and matrices.

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