FEC.cpp

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/*
 * This code appears to be mostly copied from with some additional minor
 * code provided by Michael Thompson.
 *
 *  http://www.piclist.com/techref/method/error/rs-gp-pk-uoh-199609/rs_c.htm
 *
 * This code claims to be written by Phil Karns in 1996.  The code at that
 * site unfortunately has no license in it.
 *
 * The most current version of Phil's library (fec-3.0.1) is at:
 *
 *   http://www.ka9q.net/code/fec/
 *
 * The fec-3.0.1 library (August 7, 2007) is distributed under LPGL.
 * 
 * By the time I got the the code, it appears that Michael Thompson had
 * attached a GNU Public License to it all.  Given that Phil Karn is
 * distributing his latest code under the GNU LGPL, my assumption is
 * that his older code is actually distributed under LGPL.  So I have
 * removed the GPL and I have reattached the LGPL back onto this code
 * (see end of this file.)
 *
 * Wayne C. Gramlich
 * September 17, 2010
 */

/*
    This code implements general purpose Reed-Solomon forward error correction
    algorithm for buffers of 3-bit to 8-bit symbols.  It is a based on code by
    Phil Karn, which is in turn a rewrite of code by Robert Morelos-Zaragoza
    and Hari Thirumoorthy, which was in turn based on an earlier program by
    Simon Rockliff.  Further information on Phil Karn's implementation can be
    found at:

    http://www.piclist.com/techref/method/error/rs-gp-pk-uoh-199609/index.htm

    This code is woefully undocumented and uncommented.  Hopefully someone
    who can understand the math within the Reed Solomon algorithm can comment
    and optimize this code further.
*/

#include <stdlib.h>
#include <alloca.h>

#include "FEC.hpp"

// No legal value in index form represents zero, so
// we use nn as a special value for this purpose.
#define ALPHA_ZERO  ((uintGF) self->nn)

// Alpha exponent for the first root of the generator polynomial.
#define ALPHA_ROOT  1

// Helper macro to find minimum value.
#define MIN(a,b) (((a) < (b)) ? (a) : (b))

// Primitive polynomial tables.  See Lin & Costello, Error Control Coding
// Appendix A  and  Lee & Messerschmitt, Digital Communication p. 453.

// 1 + x + x^3
rvUint8 pp_3[4] = { 1, 1, 0, 1 };

// 1 + x + x^4
rvUint8 pp_4[5] = { 1, 1, 0, 0, 1 };

// 1 + x^2 + x^5
rvUint8 pp_5[6] = { 1, 0, 1, 0, 0, 1 };

// 1 + x + x^6
rvUint8 pp_6[7] = { 1, 1, 0, 0, 0, 0, 1 };

// 1 + x^3 + x^7
rvUint8 pp_7[8] = { 1, 0, 0, 1, 0, 0, 0, 1 };

// 1+x^2+x^3+x^4+x^8
rvUint8 pp_8[9] = { 1, 0, 1, 1, 1, 0, 0, 0, 1 };

#if 0
// The following primitive polynomial tables are currently unused, but
// copied here for future reference.

// 1+x^4+x^9
rvUint8 pp_9[10] = { 1, 0, 0, 0, 1, 0, 0, 0, 0, 1 };

// 1+x^3+x^10
rvUint8 pp_10[11] = { 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1 };

// 1+x^2+x^11
rvUint8 pp_11[12] = { 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1 };

// 1+x+x^4+x^6+x^12
rvUint8 pp_12[13] = { 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1 };

// 1+x+x^3+x^4+x^13
rvUint8 pp_13[14] = { 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1 };

// 1+x+x^6+x^10+x^14
rvUint8 pp_14[15] = { 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1 };

// 1+x+x^15
rvUint8 pp_15[16] = { 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 };

// 1+x+x^3+x^12+x^16
rvUint8 pp_16[17] = { 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1 };
#endif

// Helper macro to compute x mod nn.
#define modnn(x) ((x) % self->nn)

static void rvFec_InitGaloisField(rvFec* self, rvUint8 *pp)
// Initialize GF(2**mm) table from the irreducible polynomial p(X) in pp[0]..pp[MM - 1].
//
// The following tables are created:
//
//   index form to polynomial form alphaTo[] contains j=alpha**i;
//   polynomial form to index form indexOf[j=alpha**i] = i
//
// The representation of the elements of GF(2**mm) is either in index form,
// where the number is the power of the primitive element alpha, which is
// convenient for multiplication (add the powers modulo 2**mm-1) or in
// polynomial form, where the bits represent the coefficients of the
// polynomial representation of the number, which is the most convenient form
// for addition.  The two forms are swapped between via lookup tables.
// This leads to fairly messy looking expressions, but unfortunately, there
// is no easy alternative when working with Galois arithmetic.
{
    rvUint16 i;
    rvUint16 mask;

    // Local references so code is easier on the eyes.
    uintGF mm = (uintGF) self->mm;
    uintGF nn = (uintGF) self->nn;
    uintGF *alphaTo = self->alphaTo;
    uintGF *indexOf = self->indexOf;

    mask = 1;
    alphaTo[mm] = 0;
    for (i = 0; i < mm; i++) {
        alphaTo[i] = (uintGF) mask;
        indexOf[alphaTo[i]] = (uintGF) i;
        if (pp[i] != 0) {
            alphaTo[mm] ^= (uintGF) mask;
        }
        mask <<= 1;
    }
    indexOf[alphaTo[mm]] = mm;
    mask >>= 1;
    for (i = mm + 1; i < nn; i++) {
        if (alphaTo[i - 1] >= mask) {
            alphaTo[i] = alphaTo[mm] ^ ((alphaTo[i - 1] ^ mask) << 1);
        } else {
            alphaTo[i] = alphaTo[i - 1] << 1;
        }
        indexOf[alphaTo[i]] = (uintGF) i;
    }
    indexOf[0] = ALPHA_ZERO;
    alphaTo[nn] = 0;
}


static void
rvFec_InitPolynomial(
  rvFec* self)
{
    // Initialize the generator polynomial table.
    rvInt16 i;
    rvInt16 j;

    // Local references so code is easier on the eyes.
    uintGF *gg = self->gg;
    uintGF *alphaTo = self->alphaTo;
    uintGF *indexOf = self->indexOf;

    gg[0] = alphaTo[ALPHA_ROOT];
    gg[1] = 1;
    for (i = 2; i <= self->paritySize; i++) {
        gg[i] = 1;
        for (j = i - 1; j > 0; --j) {
            if (gg[j] != 0) {
                gg[j] = gg[j - 1] ^
                  alphaTo[modnn((indexOf[gg[j]]) + ALPHA_ROOT + i - 1)];
            } else {
                gg[j] = gg[j - 1];
            }
        }
        gg[0] = alphaTo[modnn((indexOf[gg[0]]) + ALPHA_ROOT + i - 1)];
    }

    // Convert gg[] to index form for quicker encoding.
    for (i = 0; i <= self->paritySize; i++) {
        gg[i] = indexOf[gg[i]];
    }
}


rvFec* rvFec_New(rvInt16 symbolSize, rvInt16 dataSize, rvInt16 paritySize)
// Initialize the encoder object.
{
    rvInt16 mm;
    rvInt16 nn;
    rvInt16 kk;
    rvInt16 zeroSize;
    rvInt16 blockSize;
    rvUint8* pp = NULL;
    uintGF* gg = NULL;
    uintGF* alphaTo = NULL;
    uintGF* indexOf = NULL;
    rvFec* self = NULL;

    // Set the Reed Solomon properties.
    mm = symbolSize;
    nn = ((1 << mm) - 1);
    kk = nn - paritySize;

    // The block size is the combined data and parity size.
    blockSize = dataSize + paritySize;

    // The zero size is the size of the zero padded data.
    zeroSize = nn - blockSize;

    // Sanity check block and data size.
    if ((blockSize < 1) || (blockSize > nn)) return NULL;
    if ((dataSize < 1) || (dataSize >= blockSize)) return NULL;

    // Point to primative polynomial table.
    if (symbolSize == 3) pp = pp_3;
    else if (symbolSize == 4) pp = pp_4;
    else if (symbolSize == 5) pp = pp_5;
    else if (symbolSize == 6) pp = pp_6;
    else if (symbolSize == 7) pp = pp_7;
    else if (symbolSize == 8) pp = pp_8;
    else return NULL;

    // Allocate a new ecc object and tables.
    self = (rvFec *) malloc(sizeof(rvFec));
    gg = (uintGF*) malloc(sizeof(uintGF) * (paritySize + 1));
    alphaTo = (uintGF*) malloc(sizeof(uintGF) * (nn + 1));
    indexOf = (uintGF*) malloc(sizeof(uintGF) * (nn + 1));

    // Did we allocate the object.
    if ((self != NULL) && (gg != NULL) &&
      (alphaTo != NULL) && (indexOf != NULL)) {
        // Set the Reed Solomon properties.
        self->mm = mm;
        self->nn = nn;
        self->kk = kk;

        // Set the buffer properties.
        self->zeroSize = zeroSize;
        self->dataSize = dataSize;
        self->paritySize = paritySize;
        self->blockSize = blockSize;

        // Set the allocated tables.
        self->gg = gg;
        self->alphaTo = alphaTo;
        self->indexOf = indexOf;

        // Initialize Galois Field table
        rvFec_InitGaloisField(self, pp);

        // Initialize generator polynomial.
        rvFec_InitPolynomial(self);
    } else {
        // Clean up.
        if (self) free(self);
        if (gg) free(gg);
        if (alphaTo) free(alphaTo);
        if (indexOf) free(indexOf);
    }

    return self;
}


rvInt16 rvFec_Parity(rvFec* self, rvUint8* dataBuffer, rvUint8* parityBuffer)
// Encode parity for the data buffer.  Returns 1 if success or 0 if
// unable to encode.  Encoding is done by using a feedback shift register
// with appropriate connections specified by the elements of gg[], which
// was generated by the ecc_generate_galois_field() function.
{
    rvInt16 i;
    rvInt16 j;
    uintGF feedback;

    // Local references so code is easier on the eyes.
    uintGF *gg = self->gg;
    uintGF *alphaTo = self->alphaTo;
    uintGF *indexOf = self->indexOf;

    // Zero the parity data.
    for (i = 0; i < self->paritySize; ++i) parityBuffer[i] = 0;

    // Loop over the data portion of the encode buffer.
    for (i = self->zeroSize + self->dataSize; i >= 0; --i) {
        // Pad data with zero to account for shortened buffer when
        // determining feedback.
        feedback =
          indexOf[((i >= self->zeroSize) ? dataBuffer[i - self->zeroSize] : 0) ^
          parityBuffer[self->paritySize - 1]];

        // Is the feedback term non-zero?
        if (feedback != ALPHA_ZERO) {
            for (j = self->paritySize - 1; j > 0; --j) {
                if (gg[j] != ALPHA_ZERO) {
                    parityBuffer[j] =
                      parityBuffer[j - 1] ^ alphaTo[modnn(gg[j] + feedback)];
                } else {
                    parityBuffer[j] = parityBuffer[j - 1];
                }
            }
            parityBuffer[0] = alphaTo[modnn(gg[0] + feedback)];
        } else {
            for (j = self->paritySize - 1; j > 0; --j) {
                parityBuffer[j] = parityBuffer[j - 1];
            }
            parityBuffer[0] = 0;
        }
    }

    return 1;
}


rvInt16 rvFec_Correct(rvFec* self, rvUint8* blockBuffer)
// Correct the block buffer.  Returns 1 if success or 0 if unable to correct.
{
    rvInt16 i;
    rvInt16 j;
    rvInt16 r;
    rvInt16 el;
    rvInt16 count;
    rvInt16 syn_error;
    rvInt16 deg_omega;
    rvInt16 deg_lambda;
    rvInt16 *loc;
    uintGF q;
    uintGF *b;
    uintGF *t;
    uintGF *reg;
    uintGF *recd;
    uintGF *root;
    uintGF *omega;
    uintGF *lambda;
    uintGF *syndromes;

    // Local references so code is easier on the eyes.
    rvInt16 nn = self->nn;
    uintGF *gg = self->gg;
    uintGF *alphaTo = self->alphaTo;
    uintGF *indexOf = self->indexOf;

    // Allocate the recieve buffer off the stack.
    recd = (uintGF*) alloca(sizeof(uintGF) * nn);

    // Copy and convert the encoded buffer from polynomial form to index form.
    for (i = nn - 1; i >= 0; --i) {
        // Pad data with zero to account for shortened buffer.
        recd[i] =
          indexOf[(i >= self->zeroSize) ? blockBuffer[i - self->zeroSize] : 0];
    }

    // Allocate the syndromes buffer off the stack.
    syndromes = (uintGF*) alloca(sizeof(uintGF) * (self->paritySize + 1));

    // Initialize the syndrome error flag.
    syn_error = 0;

    // Form and store each syndrome by evaluating recd(x) at roots of g(x)
    // namely @**(ALPHA_ROOT+i), i = 0, ... ,(nn-KK-1)
    for (i = 1; i <= self->paritySize; i++) {
        uintGF syndrome = 0;

        // Loop over each data byte.
        for (j = 0; j < nn; j++) {
            // All non-zero receive values added to the syndrome.
            if (recd[j] != ALPHA_ZERO) {
                syndrome ^= alphaTo[modnn(recd[j] + (ALPHA_ROOT + i - 1) * j)];
            }
        }

        // Set flag if non-zero syndrome.
        syn_error |= syndrome;

        // Store syndrome in index form.
        syndromes[i] = indexOf[syndrome];
    }

    if (!syn_error) {
        // If all syndromes are zero there are no errors to correct.
        return 1;
    }

    // Allocate the lambda buffer off the stack.
    lambda = (uintGF*) alloca(sizeof(uintGF) * (self->paritySize + 1));

    // Clear the lampda buffer.
    for (i = 1; i < self->paritySize + 1; ++i) lambda[i] = 0;

    lambda[0] = 1;

    b = (uintGF*) alloca(sizeof(uintGF) * (self->paritySize + 1));
    t = (uintGF*) alloca(sizeof(uintGF) * (self->paritySize + 1));

    for (i = 0; i < self->paritySize + 1; i++) {
        b[i] = indexOf[lambda[i]];
    }

    // Begin Berlekamp-Massey algorithm to determine error locator polynomial.
    el = 0;

    for (r = 1; r <= self->paritySize; ++r) {
        uintGF discrepancy = 0;

        // Compute discrepancy at the r-th step in poly-form.
        for (i = 0; i < r; i++) {
            if ((lambda[i] != 0) && (syndromes[r - i] != ALPHA_ZERO)) {
                discrepancy ^=
                  alphaTo[modnn(indexOf[lambda[i]] + syndromes[r - i])];
            }
        }

        // Convert the discrepancy to index form.
        discrepancy = indexOf[discrepancy];

        // Is the discrepancy zero?
        if (discrepancy == ALPHA_ZERO) {
            // Move all array elements up.
            for (i = self->paritySize - 1; i >= 0; --i) {
                b[i + 1] = b[i];
            }

            // The first array element is zero.
            b[0] = ALPHA_ZERO;
        } else {
            // 7 lines below: T(x) <-- lambda(x) - discrepancy*x*b(x)
            t[0] = lambda[0];

            for (i = 0 ; i < self->paritySize; i++) {
                if (b[i] != ALPHA_ZERO) {
                    t[i+1] = lambda[i+1] ^ alphaTo[modnn(discrepancy + b[i])];
                } else {
                    t[i+1] = lambda[i+1];
                }
            }

            if ((2 * el) <= (r - 1)) {
                el = r - el;

                // 2 lines below: B(x) <-- inv(discrepancy) * lambda(x)
                for (i = 0; i <= self->paritySize; i++) {
                    b[i] = (lambda[i] == 0) ? ALPHA_ZERO :
                      modnn(indexOf[lambda[i]] - discrepancy + nn);
                }
            } else {
                // Move all array elements up.
                for (i = self->paritySize - 1; i >= 0; --i) {
                    b[i + 1] = b[i];
                }

                // The first array element is zero.
                b[0] = ALPHA_ZERO;
            }

            for (i = 0; i < self->paritySize + 1; ++i) lambda[i] = t[i];
        }
    }

    deg_lambda = 0;

    // Convert lambda to index form and compute deg(lambda(x)).
    for (i = 0; i < self->paritySize + 1; ++i) {
        lambda[i] = indexOf[lambda[i]];

        if (lambda[i] != ALPHA_ZERO) {
            deg_lambda = i;
        }
    }

    loc = (rvInt16*) alloca(sizeof(rvInt16) * self->paritySize);
    reg = (uintGF*) alloca(sizeof(uintGF) * (self->paritySize + 1));
    root = (uintGF*) alloca(sizeof(uintGF) * self->paritySize);

    // Find roots of the error locator polynomial by Chien search.
    for (i = 1; i < self->paritySize + 1; ++i) {
        reg[i] = lambda[i];
    }

    // Number of roots of lambda(x).
    count = 0;

    for (i = 1; i <= nn; i++) {
        q = 1;

        for (j = deg_lambda; j > 0; j--) {
            if (reg[j] != ALPHA_ZERO) {
                reg[j] = modnn(reg[j] + j);
                q ^= alphaTo[reg[j]];
            }
        }

        if (!q) {
            // Store root (index-form) and error location number.
            root[count] = (uintGF) i;
            loc[count] = nn - i;
            count++;
        }
    }

#ifdef DEBUG
    printf("\n Final error positions:\t");
    for (i = 0; i < count; i++) {
        printf("%d ", loc[i]);
    }
    printf("\n");
#endif

    // If deg(lambda) unequal to number of roots then uncorrectable error
    // detected:
    if (deg_lambda != count) {
        return 0;
    }

    deg_omega = 0;

    omega = (uintGF*) alloca(sizeof(uintGF) * (self->paritySize + 1));

    // Compute error evaluator poly omega(x) =
    // s(x) * lambda(x) (modulo x**(NN-KK))
    // in index form. Also find deg(omega).
    for (i = 0; i < self->paritySize; ++i) {
        uintGF tmp = 0;

        for (j = (deg_lambda < i) ? deg_lambda : i; j >= 0; --j) {
            if ((syndromes[i + 1 - j] != ALPHA_ZERO) &&
              (lambda[j] != ALPHA_ZERO)) {
                tmp ^= alphaTo[modnn(syndromes[i + 1 - j] + lambda[j])];
            }
        }

        if (tmp != 0) {
            deg_omega = i;
        }

        omega[i] = indexOf[tmp];
    }

    omega[self->paritySize] = ALPHA_ZERO;

    // Compute error values in poly-form.
    for (j = count - 1; j >= 0; j--) {
        uintGF den;
        uintGF num1;
        uintGF num2;

        num1 = 0;

        for (i = deg_omega; i >= 0; i--) {
            if (omega[i] != ALPHA_ZERO) {
                num1 ^= alphaTo[modnn(omega[i] + i * root[j])];
            }
        }

        num2 = alphaTo[modnn(root[j] * (ALPHA_ROOT - 1) + nn)];

        den = 0;

        // lambda[i+1] for i even is the formal derivative lambda_pr
        // of lambda[i]:
        for (i = MIN(deg_lambda, self->paritySize - 1) & ~1; i >= 0; i -=2) {
            if (lambda[i + 1] != ALPHA_ZERO) {
                den ^= alphaTo[modnn(lambda[i+1] + i * root[j])];
            }
        }

        if (den == 0) {
#ifdef DEBUG
            printf("\n ERROR: denominator = 0\n");
#endif
            return 0;
        }

        // Apply error to data.
        if (num1 != 0) {
            // We should never need to correct in the zero padding.
            if (loc[j] < self->zeroSize) {
                return 0;
            }

            // Account for shortened data.
            if ((loc[j] >= self->zeroSize) &&
              (loc[j] < self->zeroSize + self->dataSize)) {
                blockBuffer[loc[j] - self->zeroSize] ^=
                  alphaTo[modnn(indexOf[num1] + indexOf[num2] +
                  nn - indexOf[den])];
            }
        }
    }

    return 1;
}

bool
FEC__correct(
    FEC fec,
    unsigned int *data,
    unsigned int size)
{
    unsigned char data_bytes[8];
    unsigned int index;
    bool result;

    /* Load {data} into {data_bytes}: */
    for (index = 0; index < size; index++) {
        data_bytes[index] = data[index] & 0xff;
    }    

    /* Correct any errors: */
    result = rvFec_Correct(fec, data_bytes);

    /* Load {data_bytes} into {data}: */
    for (index = 0; index < size; index++) {
        data[index] = data_bytes[index];
    }    

    return result;
}

void
FEC__parity(
  FEC fec,
  unsigned int *data,
  unsigned int size)
{
    unsigned char data_bytes[8];
    unsigned int index;

    /* Load {data} into {data_bytes}: */
    for (index = 0; index < 8; index++) {
        data_bytes[index] = data[index] & 0xff;
    }    

    /* Compute the parity information: */
    rvFec_Parity(fec, data_bytes, data_bytes + 4);

    /* Load {data_bytes} into {data}: */
    for (index = 0; index < 8; index++) {
        data[index] = data_bytes[index];
    }    
}

FEC
FEC__create(
  unsigned int symbol_size,
  unsigned int data_size,
  unsigned int parity_size)
{
    return rvFec_New(symbol_size, data_size, parity_size);
}

/*

                  GNU LESSER GENERAL PUBLIC LICENSE
                       Version 2.1, February 1999

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That's all there is to it!

*/