/*
* Reed-Solomon coding and decoding
* Phil Karn (karn at ka9q.ampr.org) August 1997
*
* This file is derived from the program "new_rs_erasures.c" by Robert
* Morelos-Zaragoza (robert at spectra.eng.hawaii.edu) and Hari Thirumoorthy
* (harit at spectra.eng.hawaii.edu), Aug 1995
*
* This version is hard-wired to encode and decode the (255,223) code
* over GF(256). It contains calls to i386 assembler routines optimized
* for the Pentium.
*/
#include <stdio.h>
#include "rs32.h"
/* Primitive polynomials - see Lin & Costello, Error Control Coding Appendix A,
* and Lee & Messerschmitt, Digital Communication p. 453.
*/
/* 1+x^2+x^3+x^4+x^8 */
int Pp[MM+1] = { 1, 0, 1, 1, 1, 0, 0, 0, 1 };
/* Alpha exponent for the first root of the generator polynomial */
#define B0 1
/* index->polynomial form conversion table */
gf Alpha_to[NN + 1];
/* Polynomial->index form conversion table */
gf Index_of[NN + 1];
/* No legal value in index form represents zero, so
* we need a special value for this purpose
*/
#define A0 (NN)
/* Generator polynomial g(x)
* Degree of g(x) = 2*TT
* has roots @**B0, @**(B0+1), ... ,@^(B0+2*TT-1)
*/
gf Gg[NN - KK + 1];
/* Lookup table for multiplying by the 32 generator polynomial terms,
* packed 4 terms to each 32-bit word
*/
unsigned long Gtab[8][256];
/* Lookup table for GF multiplication
* Mtab[i][j] = j * alpha^(B0+i) (note limited range of i)
*/
unsigned char Mtab[NN-KK+1][NN+1];
/* Compute x % NN, where NN is 2**MM - 1,
* without a slow divide
*/
static inline gf
modnn(int x)
{
while (x >= NN) {
x -= NN;
x = (x >> MM) + (x & NN);
}
return x;
}
#define min(a,b) ((a) < (b) ? (a) : (b))
#define CLEAR(a,n) {\
int ci;\
for(ci=(n)-1;ci >=0;ci--)\
(a)[ci] = 0;\
}
#define COPY(a,b,n) {\
int ci;\
for(ci=(n)-1;ci >=0;ci--)\
(a)[ci] = (b)[ci];\
}
#define COPYDOWN(a,b,n) {\
int ci;\
for(ci=(n)-1;ci >=0;ci--)\
(a)[ci] = (b)[ci];\
}
/* generate GF(2**m) from the irreducible polynomial p(X) in p[0]..p[m]
lookup tables: index->polynomial form alpha_to[] contains j=alpha**i;
polynomial form -> index form index_of[j=alpha**i] = i
alpha=2 is the primitive element of GF(2**m)
HARI's COMMENT: (4/13/94) alpha_to[] can be used as follows:
Let @ represent the primitive element commonly called "alpha" that
is the root of the primitive polynomial p(x). Then in GF(2^m), for any
0 <= i <= 2^m-2,
@^i = a(0) + a(1) @ + a(2) @^2 + ... + a(m-1) @^(m-1)
where the binary vector (a(0),a(1),a(2),...,a(m-1)) is the representation
of the integer "alpha_to[i]" with a(0) being the LSB and a(m-1) the MSB. Thus for
example the polynomial representation of @^5 would be given by the binary
representation of the integer "alpha_to[5]".
Similarily, index_of[] can be used as follows:
As above, let @ represent the primitive element of GF(2^m) that is
the root of the primitive polynomial p(x). In order to find the power
of @ (alpha) that has the polynomial representation
a(0) + a(1) @ + a(2) @^2 + ... + a(m-1) @^(m-1)
we consider the integer "i" whose binary representation with a(0) being LSB
and a(m-1) MSB is (a(0),a(1),...,a(m-1)) and locate the entry
"index_of[i]". Now, @^index_of[i] is that element whose polynomial
representation is (a(0),a(1),a(2),...,a(m-1)).
NOTE:
The element alpha_to[2^m-1] = 0 always signifying that the
representation of "@^infinity" = 0 is (0,0,0,...,0).
Similarily, the element index_of[0] = A0 always signifying
that the power of alpha which has the polynomial representation
(0,0,...,0) is "infinity".
*/
static void
generate_gf(void)
{
register int i, mask;
mask = 1;
Alpha_to[MM] = 0;
for (i = 0; i < MM; i++) {
Alpha_to[i] = mask;
Index_of[Alpha_to[i]] = i;
/* If Pp[i] == 1 then, term @^i occurs in poly-repr of @^MM */
if (Pp[i] != 0)
Alpha_to[MM] ^= mask; /* Bit-wise EXOR operation */
mask <<= 1; /* single left-shift */
}
Index_of[Alpha_to[MM]] = MM;
/*
* Have obtained poly-repr of @^MM. Poly-repr of @^(i+1) is given by
* poly-repr of @^i shifted left one-bit and accounting for any @^MM
* term that may occur when poly-repr of @^i is shifted.
*/
mask >>= 1;
for (i = MM + 1; i < NN; i++) {
if (Alpha_to[i - 1] >= mask)
Alpha_to[i] = Alpha_to[MM] ^ ((Alpha_to[i - 1] ^ mask) << 1);
else
Alpha_to[i] = Alpha_to[i - 1] << 1;
Index_of[Alpha_to[i]] = i;
}
Index_of[0] = A0;
Alpha_to[NN] = 0;
}
/*
* Obtain the generator polynomial of the TT-error correcting, length
* NN=(2**MM -1) Reed Solomon code from the product of (X+@**(B0+i)), i = 0,
* ... ,(2*TT-1)
*
* Examples:
*
* If B0 = 1, TT = 1. deg(g(x)) = 2*TT = 2.
* g(x) = (x+@) (x+@**2)
*
* If B0 = 0, TT = 2. deg(g(x)) = 2*TT = 4.
* g(x) = (x+1) (x+@) (x+@**2) (x+@**3)
*/
static void
gen_poly(void)
{
register int i, j;
Gg[0] = Alpha_to[B0];
Gg[1] = 1; /* g(x) = (X+@**B0) initially */
for (i = 2; i <= NN - KK; i++) {
Gg[i] = 1;
/*
* Below multiply (Gg[0]+Gg[1]*x + ... +Gg[i]x^i) by
* (@**(B0+i-1) + x)
*/
for (j = i - 1; j > 0; j--)
if (Gg[j] != 0)
Gg[j] = Gg[j - 1] ^ Alpha_to[modnn((Index_of[Gg[j]]) + B0 + i - 1)];
else
Gg[j] = Gg[j - 1];
/* Gg[0] can never be zero */
Gg[0] = Alpha_to[modnn((Index_of[Gg[0]]) + B0 + i - 1)];
}
/* convert Gg[] to index form for quicker encoding */
for (i = 0; i <= NN - KK; i++)
Gg[i] = Index_of[Gg[i]];
}
/* Generate lookup table for fast encoding
* Each table entry contains four packed result per 32-bit word
*/
static void
gen_gtab(void)
{
int i,ii,j,k,val;
memset(Gtab,0,sizeof(Gtab));
for(i=1;i<256;i++){
ii = Index_of[i];
for(j=0;j<8;j++){
for(k=3;k>=0;k--){
val = (Gg[4*j+k] == A0) ? 0 : Alpha_to[modnn(ii+Gg[4*j+k])];
Gtab[j][i] = (Gtab[j][i] << 8) | val;
}
}
}
}
/* Generate lookup table for fast syndrome computation in the decoder
* Given a field element x in polynomial form, the table generated here
* when indexed as Mtab[i][x] gives x * alpha**(B0+i)
*/
static void
gen_mtab(void)
{
int i,j;
for(i=0;i<NN-KK;i++){
Mtab[i][0] = 0;
for(j=1;j<=NN;j++){
Mtab[i][j] = Alpha_to[modnn(Index_of[j] + B0 + i)];
}
}
}
void
init_rs(void)
{
generate_gf();
gen_poly();
gen_gtab();
gen_mtab();
}
/* Errors + erasures decoding of (255,223) RS code */
int
rsd32(dtype data[NN], int eras_pos[NN-KK], int no_eras)
{
int deg_lambda, el, deg_omega;
int i, j, r;
gf u,q,tmp,num1,num2,den,discr_r;
gf lambda[NN-KK + 1], s[NN-KK + 1]; /* Err+Eras Locator poly
* and syndrome poly */
gf b[NN-KK + 1], t[NN-KK + 1], omega[NN-KK + 1];
gf root[NN-KK], reg[NN-KK + 1];
int syn_error, count;
unsigned char st[NN-KK];
/*#define noasm */
#ifdef noasm
memset(st,0,sizeof(st));
for(i=254;i>=0;i--)
for(j=0;j<32;j++)
st[j] = data[i] ^ Mtab[j][st[j]];
#else
rssyndrome(data,st);
#endif
/* Convert syndromes to index form, checking for nonzero condition */
syn_error = 0;
for(i=0;i<NN-KK;i++){
syn_error |= st[i];
s[i+1] = Index_of[st[i]];
}
if (!syn_error) {
/*
* if syndrome is zero, data[] is a codeword and there are no
* errors to correct. So return data[] unmodified
*/
return 0;
}
/* Check for and quickly correct the special case of a single error.
* This is indicated by
* s[i] / s[i+1] = @^-k for all i and some constant k,
* (s in polynomial form), or
* s[i] - s[i+1] = -k,
* (s in index form)
*
* See Clark & Cain, "Error Correction Coding for Digital
* Communications", p209-210.
*/
if(s[2] != A0 && s[1] != A0){
tmp = modnn(s[2] - s[1] + NN); /* s[] in index form */
for(i=2;i<NN-KK;i++){
if(s[i+1] == A0 || s[i] == A0
|| modnn(s[i+1] - s[i] + NN) != tmp)
break;
}
if(i == NN-KK){
/* single error */
data[tmp] ^= Alpha_to[modnn(s[1] - tmp + NN)];
return 1;
}
}
CLEAR(&lambda[1],NN-KK);
lambda[0] = 1;
if (no_eras > 0) {
/* Init lambda to be the erasure locator polynomial */
lambda[1] = Alpha_to[eras_pos[0]];
for (i = 1; i < no_eras; i++) {
u = eras_pos[i];
for (j = i+1; j > 0; j--) {
tmp = Index_of[lambda[j - 1]];
if(tmp != A0)
lambda[j] ^= Alpha_to[modnn(u + tmp)];
}
}
}
for(i=0;i<NN-KK+1;i++)
b[i] = Index_of[lambda[i]];
/*
* Begin Berlekamp-Massey algorithm to determine error+erasure
* locator polynomial
*/
r = no_eras;
el = no_eras;
while (++r <= NN-KK) { /* r is the step number */
/* Compute discrepancy at the r-th step in poly-form */
discr_r = 0;
for (i = 0; i < r; i++){
if ((lambda[i] != 0) && (s[r - i] != A0)) {
discr_r ^= Alpha_to[modnn(Index_of[lambda[i]] + s[r - i])];
}
}
discr_r = Index_of[discr_r]; /* Index form */
if (discr_r == A0) {
/* 2 lines below: B(x) <-- x*B(x) */
COPYDOWN(&b[1],b,NN-KK);
b[0] = A0;
} else {
/* 7 lines below: T(x) <-- lambda(x) - discr_r*x*b(x) */
t[0] = lambda[0];
for (i = 0 ; i < NN-KK; 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 (i = 0; i <= NN-KK; i++)
b[i] = (lambda[i] == 0) ? A0 : modnn(Index_of[lambda[i]] - discr_r + NN);
} else {
/* 2 lines below: B(x) <-- x*B(x) */
COPYDOWN(&b[1],b,NN-KK);
b[0] = A0;
}
COPY(lambda,t,NN-KK+1);
}
}
/* Convert lambda to index form and compute deg(lambda(x)) */
deg_lambda = 0;
for(i=0;i<NN-KK+1;i++){
lambda[i] = Index_of[lambda[i]];
if(lambda[i] != A0)
deg_lambda = i;
}
/*
* Find roots of the error+erasure locator polynomial. By Chien
* Search
*/
COPY(®[1],&lambda[1],NN-KK);
count = 0; /* Number of roots of lambda(x) */
for (i = 1; i <= NN; i++) {
q = 1;
for (j = deg_lambda; j > 0; j--)
if (reg[j] != A0) {
reg[j] = modnn(reg[j] + j);
q ^= Alpha_to[reg[j]];
}
if (!q) {
/* store root (index-form) and error location number */
root[count++] = i;
if(count == deg_lambda)
/* Found all possible roots, no point in continuing */
break;
}
}
#ifdef DEBUG
printf("\n Final error positions:\t");
for (i = 0; i < count; i++)
printf("%d ", NN - root[i]);
printf("\n");
#endif
if (deg_lambda != count) {
/*
* deg(lambda) unequal to number of roots => uncorrectable
* error detected
*/
return -1;
}
/*
* Compute err+eras evaluator poly omega(x) = s(x)*lambda(x) (modulo
* x**(NN-KK)). in index form. Also find deg(omega).
*/
deg_omega = 0;
for (i = 0; i < NN-KK;i++){
tmp = 0;
j = (deg_lambda < i) ? deg_lambda : i;
for(;j >= 0; j--){
if ((s[i + 1 - j] != A0) && (lambda[j] != A0))
tmp ^= Alpha_to[modnn(s[i + 1 - j] + lambda[j])];
}
if(tmp != 0)
deg_omega = i;
omega[i] = Index_of[tmp];
}
omega[NN-KK] = A0;
/*
* Compute error values in poly-form. num1 = omega(inv(X(l))), num2 =
* inv(X(l))**(B0-1) and den = lambda_pr(inv(X(l))) all in poly-form
*/
for (j = count-1; j >=0; j--) {
num1 = 0;
for (i = deg_omega; i >= 0; i--) {
if (omega[i] != A0)
num1 ^= Alpha_to[modnn(omega[i] + i * root[j])];
}
num2 = Alpha_to[modnn(root[j] * (B0 - 1) + NN)];
den = 0;
/* lambda[i+1] for i even is the formal derivative lambda_pr of lambda[i] */
for (i = min(deg_lambda,NN-KK-1) & ~1; i >= 0; i -=2) {
if(lambda[i+1] != A0)
den ^= Alpha_to[modnn(lambda[i+1] + i * root[j])];
}
if (den == 0) {
#ifdef DEBUG
printf("\n ERROR: denominator = 0\n");
#endif
return -1;
}
/* Apply error to data */
if (num1 != 0) {
data[NN - root[j]] ^= Alpha_to[modnn(Index_of[num1] + Index_of[num2] + NN - Index_of[den])];
}
}
return count;
}