465 lines
14 KiB
C++
465 lines
14 KiB
C++
#ifndef REEDSOLOMON_HPP_b1405fdab6374ba2a4e65e8d45ec3d80
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#define REEDSOLOMON_HPP_b1405fdab6374ba2a4e65e8d45ec3d80
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/**
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* Code taken and adapted from www.eccpage.com/rs.c
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* Credit goes to Mr Simon Rockliff.
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*
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* Tried before with the implementation from ITPP library but couldn't make it produce the same outputs
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* expected from the P25 transmissions that I have tested. This implementation does work.
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*/
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/* This program is an encoder/decoder for Reed-Solomon codes. Encoding is in
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systematic form, decoding via the Berlekamp iterative algorithm.
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In the present form , the constants mm, nn, tt, and kk=nn-2tt must be
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specified (the double letters are used simply to avoid clashes with
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other n,k,t used in other programs into which this was incorporated!)
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Also, the irreducible polynomial used to generate GF(2**mm) must also be
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entered -- these can be found in Lin and Costello, and also Clark and Cain.
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The representation of the elements of GF(2**m) is either in index form,
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where the number is the power of the primitive element alpha, which is
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convenient for multiplication (add the powers modulo 2**m-1) or in
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polynomial form, where the bits represent the coefficients of the
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polynomial representation of the number, which is the most convenient form
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for addition. The two forms are swapped between via lookup tables.
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This leads to fairly messy looking expressions, but unfortunately, there
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is no easy alternative when working with Galois arithmetic.
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The code is not written in the most elegant way, but to the best
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of my knowledge, (no absolute guarantees!), it works.
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However, when including it into a simulation program, you may want to do
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some conversion of global variables (used here because I am lazy!) to
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local variables where appropriate, and passing parameters (eg array
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addresses) to the functions may be a sensible move to reduce the number
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of global variables and thus decrease the chance of a bug being introduced.
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This program does not handle erasures at present, but should not be hard
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to adapt to do this, as it is just an adjustment to the Berlekamp-Massey
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algorithm. It also does not attempt to decode past the BCH bound -- see
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Blahut "Theory and practice of error control codes" for how to do this.
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Simon Rockliff, University of Adelaide 21/9/89
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26/6/91 Slight modifications to remove a compiler dependent bug which hadn't
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previously surfaced. A few extra comments added for clarity.
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Appears to all work fine, ready for posting to net!
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Notice
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--------
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This program may be freely modified and/or given to whoever wants it.
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A condition of such distribution is that the author's contribution be
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acknowledged by his name being left in the comments heading the program,
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however no responsibility is accepted for any financial or other loss which
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may result from some unforseen errors or malfunctioning of the program
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during use.
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Simon Rockliff, 26th June 1991
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*/
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#include <cmath>
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template<int TT> class CReedSolomon63
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{
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private:
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static const int MM = 6; /* RS code over GF(2**mm) */
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static const int NN = 63; /* nn=2**mm -1 length of codeword */
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//int tt; /* number of errors that can be corrected */
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//int kk; /* kk = nn-2*tt */
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static const int KK = NN - 2 * TT;
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// distance = nn-kk+1 = 2*tt+1
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int* alpha_to;
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int* index_of;
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int* gg;
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void generate_gf(int* generator_polinomial)
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/* generate GF(2**mm) from the irreducible polynomial p(X) in pp[0]..pp[mm]
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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**mm)
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*/
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{
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register int i, mask;
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mask = 1;
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alpha_to[MM] = 0;
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for (i = 0; i < MM; i++) {
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alpha_to[i] = mask;
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index_of[alpha_to[i]] = i;
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if (generator_polinomial[i] != 0)
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alpha_to[MM] ^= mask;
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mask <<= 1;
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}
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index_of[alpha_to[MM]] = MM;
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mask >>= 1;
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for (i = MM + 1; i < NN; i++) {
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if (alpha_to[i - 1] >= mask)
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alpha_to[i] = alpha_to[MM] ^ ((alpha_to[i - 1] ^ mask) << 1);
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else
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alpha_to[i] = alpha_to[i - 1] << 1;
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index_of[alpha_to[i]] = i;
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}
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index_of[0] = -1;
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}
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void gen_poly()
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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+alpha**i), i=1..2*tt
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*/
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{
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register int i, j;
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gg[0] = 2; /* primitive element alpha = 2 for GF(2**mm) */
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gg[1] = 1; /* g(x) = (X+alpha) initially */
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for (i = 2; i <= NN - KK; i++) {
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gg[i] = 1;
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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[(index_of[gg[j]] + i) % NN];
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else
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gg[j] = gg[j - 1];
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gg[0] = alpha_to[(index_of[gg[0]] + i) % NN]; /* gg[0] can never be zero */
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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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public:
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CReedSolomon63()
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{
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alpha_to = new int[NN + 1];
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index_of = new int[NN + 1];
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gg = new int[NN - KK + 1];
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// Polynom used in P25 is alpha**6+alpha+1
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int generator_polinomial[] = { 1, 1, 0, 0, 0, 0, 1 }; /* specify irreducible polynomial coeffts */
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generate_gf(generator_polinomial);
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gen_poly();
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}
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virtual ~CReedSolomon63()
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{
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delete[] gg;
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delete[] index_of;
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delete[] alpha_to;
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}
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void encode(const int* data, int* bb)
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/* take the string of symbols in data[i], i=0..(k-1) and encode systematically
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to produce 2*tt parity symbols in bb[0]..bb[2*tt-1]
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data[] is input and bb[] is output in polynomial form.
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Encoding is done by using a feedback shift register with appropriate
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connections specified by the elements of gg[], which was generated above.
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Codeword is c(X) = data(X)*X**(nn-kk)+ b(X) */
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{
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register int i, j;
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int feedback;
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for (i = 0; i < NN - KK; i++)
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bb[i] = 0;
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for (i = KK - 1; i >= 0; i--) {
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feedback = index_of[data[i] ^ bb[NN - KK - 1]];
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if (feedback != -1) {
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for (j = NN - KK - 1; j > 0; j--)
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if (gg[j] != -1)
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bb[j] = bb[j - 1] ^ alpha_to[(gg[j] + feedback) % NN];
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else
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bb[j] = bb[j - 1];
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bb[0] = alpha_to[(gg[0] + feedback) % NN];
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}
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else {
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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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}
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int decode(const int* input, int* recd)
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/* assume we have received bits grouped into mm-bit symbols in recd[i],
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i=0..(nn-1), and recd[i] is polynomial form.
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We first compute the 2*tt syndromes by substituting alpha**i into rec(X) and
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evaluating, storing the syndromes in s[i], i=1..2tt (leave s[0] zero) .
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Then we use the Berlekamp iteration to find the error location polynomial
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elp[i]. If the degree of the elp is >tt, we cannot correct all the errors
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and hence just put out the information symbols uncorrected. If the degree of
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elp is <=tt, we substitute alpha**i , i=1..n into the elp to get the roots,
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hence the inverse roots, the error location numbers. If the number of errors
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located does not equal the degree of the elp, we have more than tt errors
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and cannot correct them. Otherwise, we then solve for the error value at
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the error location and correct the error. The procedure is that found in
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Lin and Costello. For the cases where the number of errors is known to be too
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large to correct, the information symbols as received are output (the
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advantage of systematic encoding is that hopefully some of the information
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symbols will be okay and that if we are in luck, the errors are in the
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parity part of the transmitted codeword). Of course, these insoluble cases
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can be returned as error flags to the calling routine if desired. */
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{
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register int i, j, u, q;
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int elp[NN - KK + 2][NN - KK], d[NN - KK + 2], l[NN - KK + 2], u_lu[NN - KK
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+ 2], s[NN - KK + 1];
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int count = 0, syn_error = 0, root[TT], loc[TT], z[TT + 1], err[NN], reg[TT
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+ 1];
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int irrecoverable_error = 0;
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for (int i = 0; i < NN; i++)
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recd[i] = index_of[input[i]]; /* put recd[i] into index form (ie as powers of alpha) */
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/* first form the syndromes */
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for (i = 1; i <= NN - KK; i++) {
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s[i] = 0;
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for (j = 0; j < NN; j++)
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if (recd[j] != -1)
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s[i] ^= alpha_to[(recd[j] + i * j) % NN]; /* recd[j] in index form */
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/* convert syndrome from polynomial form to index form */
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if (s[i] != 0)
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syn_error = 1; /* set flag if non-zero syndrome => error */
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s[i] = index_of[s[i]];
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}
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if (syn_error) /* if errors, try and correct */
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{
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/* compute the error location polynomial via the Berlekamp iterative algorithm,
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following the terminology of Lin and Costello : d[u] is the 'mu'th
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discrepancy, where u='mu'+1 and 'mu' (the Greek letter!) is the step number
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ranging from -1 to 2*tt (see L&C), l[u] is the
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degree of the elp at that step, and u_l[u] is the difference between the
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step number and the degree of the elp.
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*/
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/* initialise table entries */
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d[0] = 0; /* index form */
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d[1] = s[1]; /* index form */
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elp[0][0] = 0; /* index form */
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elp[1][0] = 1; /* polynomial form */
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for (i = 1; i < NN - KK; i++) {
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elp[0][i] = -1; /* index form */
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elp[1][i] = 0; /* polynomial form */
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}
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l[0] = 0;
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l[1] = 0;
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u_lu[0] = -1;
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u_lu[1] = 0;
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u = 0;
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do {
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u++;
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if (d[u] == -1) {
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l[u + 1] = l[u];
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for (i = 0; i <= l[u]; i++) {
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elp[u + 1][i] = elp[u][i];
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elp[u][i] = index_of[elp[u][i]];
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}
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}
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else
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/* search for words with greatest u_lu[q] for which d[q]!=0 */
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{
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q = u - 1;
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while ((d[q] == -1) && (q > 0))
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q--;
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/* have found first non-zero d[q] */
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if (q > 0) {
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j = q;
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do {
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j--;
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if ((d[j] != -1) && (u_lu[q] < u_lu[j]))
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q = j;
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} while (j > 0);
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};
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/* have now found q such that d[u]!=0 and u_lu[q] is maximum */
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/* store degree of new elp polynomial */
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if (l[u] > l[q] + u - q)
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l[u + 1] = l[u];
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else
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l[u + 1] = l[q] + u - q;
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/* form new elp(x) */
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for (i = 0; i < NN - KK; i++)
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elp[u + 1][i] = 0;
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for (i = 0; i <= l[q]; i++)
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if (elp[q][i] != -1)
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elp[u + 1][i + u - q] = alpha_to[(d[u] + NN - d[q]
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+ elp[q][i]) % NN];
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for (i = 0; i <= l[u]; i++) {
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elp[u + 1][i] ^= elp[u][i];
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elp[u][i] = index_of[elp[u][i]]; /*convert old elp value to index*/
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}
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}
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u_lu[u + 1] = u - l[u + 1];
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/* form (u+1)th discrepancy */
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if (u < NN - KK) /* no discrepancy computed on last iteration */
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{
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if (s[u + 1] != -1)
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d[u + 1] = alpha_to[s[u + 1]];
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else
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d[u + 1] = 0;
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for (i = 1; i <= l[u + 1]; i++)
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if ((s[u + 1 - i] != -1) && (elp[u + 1][i] != 0))
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d[u + 1] ^= alpha_to[(s[u + 1 - i]
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+ index_of[elp[u + 1][i]]) % NN];
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d[u + 1] = index_of[d[u + 1]]; /* put d[u+1] into index form */
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}
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} while ((u < NN - KK) && (l[u + 1] <= TT));
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u++;
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if (l[u] <= TT) /* can correct error */
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{
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/* put elp into index form */
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for (i = 0; i <= l[u]; i++)
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elp[u][i] = index_of[elp[u][i]];
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/* find roots of the error location polynomial */
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for (i = 1; i <= l[u]; i++)
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reg[i] = elp[u][i];
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count = 0;
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for (i = 1; i <= NN; i++) {
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q = 1;
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for (j = 1; j <= l[u]; j++)
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if (reg[j] != -1) {
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reg[j] = (reg[j] + j) % NN;
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q ^= alpha_to[reg[j]];
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};
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if (!q) /* store root and error location 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 == l[u]) /* no. roots = degree of elp hence <= tt errors */
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{
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/* form polynomial z(x) */
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for (i = 1; i <= l[u]; i++) /* Z[0] = 1 always - do not need */
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{
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if ((s[i] != -1) && (elp[u][i] != -1))
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z[i] = alpha_to[s[i]] ^ alpha_to[elp[u][i]];
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else if ((s[i] != -1) && (elp[u][i] == -1))
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z[i] = alpha_to[s[i]];
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else if ((s[i] == -1) && (elp[u][i] != -1))
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z[i] = alpha_to[elp[u][i]];
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else
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z[i] = 0;
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for (j = 1; j < i; j++)
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if ((s[j] != -1) && (elp[u][i - j] != -1))
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z[i] ^= alpha_to[(elp[u][i - j] + s[j]) % NN];
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z[i] = index_of[z[i]]; /* put into index form */
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};
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/* evaluate errors at locations given by error location numbers loc[i] */
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for (i = 0; i < NN; i++) {
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err[i] = 0;
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if (recd[i] != -1) /* convert recd[] to polynomial form */
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recd[i] = alpha_to[recd[i]];
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else
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recd[i] = 0;
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}
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for (i = 0; i < l[u]; i++) /* compute numerator of error term first */
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{
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err[loc[i]] = 1; /* accounts for z[0] */
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for (j = 1; j <= l[u]; j++)
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if (z[j] != -1)
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err[loc[i]] ^= alpha_to[(z[j] + j * root[i]) % NN];
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if (err[loc[i]] != 0) {
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err[loc[i]] = index_of[err[loc[i]]];
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q = 0; /* form denominator of error term */
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for (j = 0; j < l[u]; j++)
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if (j != i)
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q += index_of[1
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^ alpha_to[(loc[j] + root[i]) % NN]];
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q = q % NN;
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err[loc[i]] = alpha_to[(err[loc[i]] - q + NN) % NN];
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recd[loc[i]] ^= err[loc[i]]; /*recd[i] must be in polynomial form */
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}
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}
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}
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else {
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/* no. roots != degree of elp => >tt errors and cannot solve */
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irrecoverable_error = 1;
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}
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}
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else {
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/* elp has degree >tt hence cannot solve */
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irrecoverable_error = 1;
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}
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}
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else {
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/* no non-zero syndromes => no errors: output received codeword */
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for (i = 0; i < NN; i++)
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if (recd[i] != -1) /* convert recd[] to polynomial form */
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recd[i] = alpha_to[recd[i]];
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else
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recd[i] = 0;
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}
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if (irrecoverable_error) {
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for (i = 0; i < NN; i++) /* could return error flag if desired */
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if (recd[i] != -1) /* convert recd[] to polynomial form */
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recd[i] = alpha_to[recd[i]];
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else
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recd[i] = 0; /* just output received codeword as is */
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}
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return irrecoverable_error;
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}
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};
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/**
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* Convenience class that does a Reed-Solomon (36,20,17) error correction adapting input and output to
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* the DSD data format: hex words packed as char arrays.
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*/
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class CRS362017 : public CReedSolomon63<8>
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{
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public:
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CRS362017();
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~CRS362017();
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bool decode(unsigned char* data);
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void encode(unsigned char* data);
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private:
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};
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/**
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* Convenience class that does a Reed-Solomon (24,12,13) error correction adapting input and output to
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* the DSD data format: hex words packed as char arrays.
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*/
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class CRS241213 : public CReedSolomon63<6>
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{
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public:
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CRS241213();
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~CRS241213();
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bool decode(unsigned char* data);
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void encode(unsigned char* data);
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private:
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};
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/**
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* Convenience class that does a Reed-Solomon (24,16,9) error correction adapting input and output to
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* the DSD data format: hex words packed as char arrays.
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*/
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class CRS24169 : public CReedSolomon63<4>
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{
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public:
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CRS24169();
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~CRS24169();
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bool decode(unsigned char* data);
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void encode(unsigned char* data);
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private:
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};
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#endif
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