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ECC Reed-Solomon: errors & erasures

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Zilog80 2017-11-24 02:23:13 +01:00
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ecc/bch_ecc.c
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@ -47,6 +47,7 @@ int rs_init_RS255(void);
int rs_init_BCH64(void); int rs_init_BCH64(void);
int rs_encode(ui8_t cw[]); int rs_encode(ui8_t cw[]);
int rs_decode(ui8_t cw[], ui8_t *err_pos, ui8_t *err_val); int rs_decode(ui8_t cw[], ui8_t *err_pos, ui8_t *err_val);
int rs_decode_ErrEra(ui8_t cw[], int nera, ui8_t era_pos[], ui8_t *err_pos, ui8_t *err_val);
int rs_decode_bch_gf2t2(ui8_t cw[], ui8_t *err_pos, ui8_t *err_val); int rs_decode_bch_gf2t2(ui8_t cw[], ui8_t *err_pos, ui8_t *err_val);
// --- // ---
@ -411,7 +412,7 @@ int poly_divmod(ui8_t p[], ui8_t q[], ui8_t *d, ui8_t *r) {
deg_p = poly_deg(p); deg_p = poly_deg(p);
deg_q = poly_deg(q); deg_q = poly_deg(q);
if (deg_q < 0) return -1; // DIV_BY_ZERO if (deg_q < 0) return -1; // q=0: DIV_BY_ZERO
for (i = 0; i <= MAX_DEG; i++) d[i] = 0; for (i = 0; i <= MAX_DEG; i++) d[i] = 0;
for (i = 0; i <= MAX_DEG; i++) r[i] = 0; for (i = 0; i <= MAX_DEG; i++) r[i] = 0;
@ -423,13 +424,12 @@ int poly_divmod(ui8_t p[], ui8_t q[], ui8_t *d, ui8_t *r) {
for (i = 0; i <= deg_p; i++) d[i] = GF_mul(p[i], c); for (i = 0; i <= deg_p; i++) d[i] = GF_mul(p[i], c);
for (i = 0; i <= MAX_DEG; i++) r[i] = 0; for (i = 0; i <= MAX_DEG; i++) r[i] = 0;
} }
else if (deg_p == 0) { else if (deg_p < 0) { // p=0
for (i = 0; i <= MAX_DEG; i++) { for (i = 0; i <= MAX_DEG; i++) {
d[i] = 0; d[i] = 0;
r[i] = 0; r[i] = 0;
} }
} }
else if (deg_p < deg_q) { else if (deg_p < deg_q) {
for (i = 0; i <= MAX_DEG; i++) d[i] = 0; for (i = 0; i <= MAX_DEG; i++) d[i] = 0;
for (i = 0; i <= deg_p; i++) r[i] = p[i]; // r(x)=p(x), deg(r)<deg(q) for (i = 0; i <= deg_p; i++) r[i] = p[i]; // r(x)=p(x), deg(r)<deg(q)
@ -508,10 +508,10 @@ int polyGF_eggT(int deg, ui8_t a[], ui8_t b[], // in
for (i = 0; i <= MAX_DEG; i++) { r0[i] = a[i]; } for (i = 0; i <= MAX_DEG; i++) { r0[i] = a[i]; }
for (i = 0; i <= MAX_DEG; i++) { r1[i] = b[i]; } for (i = 0; i <= MAX_DEG; i++) { r1[i] = b[i]; }
s0[0] = 1; for (i = 1; i <= MAX_DEG; i++) { s0[i] = 0; } s0[0] = 1; for (i = 1; i <= MAX_DEG; i++) { s0[i] = 0; } // s0 = 1
s1[0] = 0; for (i = 1; i <= MAX_DEG; i++) { s1[i] = 0; } s1[0] = 0; for (i = 1; i <= MAX_DEG; i++) { s1[i] = 0; } // s1 = 0
t0[0] = 0; for (i = 1; i <= MAX_DEG; i++) { t0[i] = 0; } t0[0] = 0; for (i = 1; i <= MAX_DEG; i++) { t0[i] = 0; } // t0 = 0
t1[0] = 1; for (i = 1; i <= MAX_DEG; i++) { t1[i] = 0; } t1[0] = 1; for (i = 1; i <= MAX_DEG; i++) { t1[i] = 0; } // t1 = 1
for (i = 0; i <= MAX_DEG; i++) { r2[i] = 0; } for (i = 0; i <= MAX_DEG; i++) { r2[i] = 0; }
for (i = 0; i <= MAX_DEG; i++) { s2[i] = 0; } for (i = 0; i <= MAX_DEG; i++) { s2[i] = 0; }
for (i = 0; i <= MAX_DEG; i++) { t2[i] = 0; } for (i = 0; i <= MAX_DEG; i++) { t2[i] = 0; }
@ -545,10 +545,10 @@ int polyGF_eggT(int deg, ui8_t a[], ui8_t b[], // in
} }
static static
int polyGF_lfsr(int deg, ui8_t S[], int polyGF_lfsr(int deg, int x2t, ui8_t S[],
ui8_t *Lambda, ui8_t *Omega ) { ui8_t *Lambda, ui8_t *Omega ) {
// BCH/RS/LFSR: deg=t, // BCH/RS/LFSR: deg=t+e/2, e=#erasures
// S(x)Lambda(x) = Omega(x) mod x^(2t) // S(x)Lambda(x) = Omega(x) mod x^2t
int i; int i;
ui8_t r0[MAX_DEG+1], r1[MAX_DEG+1], r2[MAX_DEG+1], ui8_t r0[MAX_DEG+1], r1[MAX_DEG+1], r2[MAX_DEG+1],
s0[MAX_DEG+1], s1[MAX_DEG+1], s2[MAX_DEG+1], s0[MAX_DEG+1], s1[MAX_DEG+1], s2[MAX_DEG+1],
@ -558,13 +558,13 @@ int polyGF_lfsr(int deg, ui8_t S[],
for (i = 0; i <= MAX_DEG; i++) { Omega[i] = 0; } for (i = 0; i <= MAX_DEG; i++) { Omega[i] = 0; }
for (i = 0; i <= MAX_DEG; i++) { r0[i] = S[i]; } for (i = 0; i <= MAX_DEG; i++) { r0[i] = S[i]; }
for (i = 0; i <= MAX_DEG; i++) { r1[i] = 0; } r1[2*deg] = 1; //x^2t for (i = 0; i <= MAX_DEG; i++) { r1[i] = 0; } r1[x2t] = 1; //x^2t
s0[0] = 1; for (i = 1; i <= MAX_DEG; i++) { s0[i] = 0; } s0[0] = 1; for (i = 1; i <= MAX_DEG; i++) { s0[i] = 0; }
s1[0] = 0; for (i = 1; i <= MAX_DEG; i++) { s1[i] = 0; } s1[0] = 0; for (i = 1; i <= MAX_DEG; i++) { s1[i] = 0; }
for (i = 0; i <= MAX_DEG; i++) { r2[i] = 0; } for (i = 0; i <= MAX_DEG; i++) { r2[i] = 0; }
for (i = 0; i <= MAX_DEG; i++) { s2[i] = 0; } for (i = 0; i <= MAX_DEG; i++) { s2[i] = 0; }
while ( poly_deg(r1) >= deg ) { while ( poly_deg(r1) >= deg ) { // deg=t+e/2
poly_divmod(r0, r1, quo, r2); poly_divmod(r0, r1, quo, r2);
for (i = 0; i <= MAX_DEG; i++) { r0[i] = r1[i]; } for (i = 0; i <= MAX_DEG; i++) { r0[i] = r1[i]; }
for (i = 0; i <= MAX_DEG; i++) { r1[i] = r2[i]; } for (i = 0; i <= MAX_DEG; i++) { r1[i] = r2[i]; }
@ -602,7 +602,7 @@ ui8_t forney(ui8_t x, ui8_t Omega[], ui8_t Lambda[]) {
// Y = X^(1-b) * Omega(X^(-1))/Lambda'(X^(-1)) // Y = X^(1-b) * Omega(X^(-1))/Lambda'(X^(-1))
poly_D(Lambda, DLam); poly_D(Lambda, DLam);
w = poly_eval(Omega, x); w = poly_eval(Omega, x);
z = poly_eval(DLam, x); z = poly_eval(DLam, x); if (z == 0) { return -00; }
Y = GF_mul(w, GF_inv(z)); Y = GF_mul(w, GF_inv(z));
if (RS.b == 0) Y = GF_mul(GF_inv(x), Y); if (RS.b == 0) Y = GF_mul(GF_inv(x), Y);
else if (RS.b > 1) { else if (RS.b > 1) {
@ -613,6 +613,44 @@ ui8_t forney(ui8_t x, ui8_t Omega[], ui8_t Lambda[]) {
return Y; return Y;
} }
static
int era_sigma(int n, ui8_t era_pos[], ui8_t *sigma) {
int i;
ui8_t Xa[MAX_DEG+1], sig[MAX_DEG+1];
ui8_t a_i;
for (i = 0; i <= MAX_DEG; i++) sig[i] = 0;
for (i = 0; i <= MAX_DEG; i++) Xa[i] = 0;
// sigma(X)=(1 - alpha^j1 X)...(1 - alpha^jn X)
// j_{i+1} = era_pos[i]
sig[0] = 1;
Xa[0] = 1;
for (i = 0; i < n; i++) { // n <= 2*RS.t
a_i = exp_a[era_pos[i] % (GF.ord-1)];
Xa[1] = a_i; // Xalp[0..1]: (1-alpha^(j_)X)
poly_mul(sig, Xa, sig);
}
for (i = 0; i <= MAX_DEG; i++) sigma[i] = sig[i];
return 0;
}
static
int syndromes(ui8_t cw[], ui8_t *S) {
int i, errors = 0;
ui8_t a_i;
// syndromes: e_j=S(alpha^(b+i))
for (i = 0; i < 2*RS.t; i++) {
a_i = exp_a[(RS.b+i) % (GF.ord-1)]; // alpha^(b+i)
S[i] = poly_eval(cw, a_i);
if (S[i]) errors = 1;
}
return errors;
}
static static
int prn_GFpoly(ui32_t p) { int prn_GFpoly(ui32_t p) {
@ -641,7 +679,6 @@ int prn_GFpoly(ui32_t p) {
return 0; return 0;
} }
static static
void prn_table(void) { void prn_table(void) {
int i; int i;
@ -673,6 +710,7 @@ void prn_table(void) {
printf("\n"); printf("\n");
} }
int rs_init_RS255() { int rs_init_RS255() {
int i, check_gen; int i, check_gen;
ui8_t Xalp[MAX_DEG+1]; ui8_t Xalp[MAX_DEG+1];
@ -711,81 +749,105 @@ int rs_init_BCH64() {
return check_gen; return check_gen;
} }
static
int syndromes(ui8_t cw[], ui8_t *S) {
int i, errors = 0;
ui8_t a_i;
// syndromes: e_j=S(alpha^(b+i))
for (i = 0; i < 2*RS.t; i++) {
a_i = exp_a[(RS.b+i) % (GF.ord-1)]; // alpha^(b+i)
S[i] = poly_eval(cw, a_i);
if (S[i]) errors = 1;
}
return errors;
}
int rs_encode(ui8_t cw[]) { int rs_encode(ui8_t cw[]) {
int j; int j;
ui8_t parity[MAX_DEG+1], ui8_t parity[MAX_DEG+1],
d[MAX_DEG+1]; d[MAX_DEG+1];
for (j = 0; j < RS.R; j++) cw[j] = 0; for (j = 0; j < RS.R; j++) cw[j] = 0;
for (j = 0; j <=MAX_DEG; j++) parity[j] = 0; for (j = 0; j <= MAX_DEG; j++) parity[j] = 0;
poly_divmod(cw, RS.g, d, parity); poly_divmod(cw, RS.g, d, parity);
//if (poly_deg(parity) >= RS.R) return -1; //if (poly_deg(parity) >= RS.R) return -1;
for (j = 0; j <= poly_deg(parity); j++) cw[j] = parity[j]; for (j = 0; j <= poly_deg(parity); j++) cw[j] = parity[j];
return 0; return 0;
} }
int rs_decode(ui8_t cw[], ui8_t *err_pos, ui8_t *err_val) { // 2*Errors + Erasure <= 2*RS.t
ui8_t x, gamma, int rs_decode_ErrEra(ui8_t cw[], int nera, ui8_t era_pos[],
S[MAX_DEG+1], ui8_t *err_pos, ui8_t *err_val) {
ui8_t x, gamma;
ui8_t S[MAX_DEG+1],
Lambda[MAX_DEG+1], Lambda[MAX_DEG+1],
Omega[MAX_DEG+1]; Omega[MAX_DEG+1],
int i, n, errors = 0; sigma[MAX_DEG+1],
sigLam[MAX_DEG+1];
int deg_sigLam, deg_Lambda, deg_Omega;
int i, nerr, errera = 0;
for (i = 0; i < RS.t; i++) { err_pos[i] = 0; } if (nera > 2*RS.t) { return -4; }
for (i = 0; i < RS.t; i++) { err_val[i] = 0; }
for (i = 0; i < 2*RS.t; i++) { err_pos[i] = 0; }
for (i = 0; i < 2*RS.t; i++) { err_val[i] = 0; }
// IF: erasures set 0
// for (i = 0; i < nera; i++) cw[era_pos[i]] = 0x00; // erasures
// THEN: restore cw[era_pos[i]], if errera < 0
for (i = 0; i <= MAX_DEG; i++) { S[i] = 0; } for (i = 0; i <= MAX_DEG; i++) { S[i] = 0; }
errors = syndromes(cw, S); errera = syndromes(cw, S);
// wenn S(x)=0 , dann poly_divmod(cw, RS.g, d, rem): rem=0 // wenn S(x)=0 , dann poly_divmod(cw, RS.g, d, rem): rem=0
if (errors) { for (i = 0; i <= MAX_DEG; i++) { sigma[i] = 0; }
polyGF_lfsr(RS.t, S, Lambda, Omega); sigma[0] = 1;
if (nera > 0) {
era_sigma(nera, era_pos, sigma);
poly_mul(sigma, S, S);
for (i = 2*RS.t; i <= MAX_DEG; i++) S[i] = 0; // S = sig*S mod x^12
}
if (errera)
{
polyGF_lfsr(RS.t+nera/2, 2*RS.t, S, Lambda, Omega);
deg_Lambda = poly_deg(Lambda);
deg_Omega = poly_deg(Omega);
if (deg_Omega >= deg_Lambda + nera) {
errera = -3;
return errera;
}
gamma = Lambda[0]; gamma = Lambda[0];
if (gamma) { if (gamma) {
for (i = poly_deg(Lambda); i >= 0; i--) Lambda[i] = GF_mul(Lambda[i], GF_inv(gamma)); for (i = deg_Lambda; i >= 0; i--) Lambda[i] = GF_mul(Lambda[i], GF_inv(gamma));
for (i = poly_deg(Omega) ; i >= 0; i--) Omega[i] = GF_mul( Omega[i], GF_inv(gamma)); for (i = deg_Omega ; i >= 0; i--) Omega[i] = GF_mul( Omega[i], GF_inv(gamma));
poly_mul(sigma, Lambda, sigLam);
deg_sigLam = poly_deg(sigLam);
} }
else { else {
errors = -2; errera = -2;
//return errors; return errera;
} }
n = 0; nerr = 0; // Errors + Erasures (erasure-pos bereits bekannt)
for (i = 1; i < GF.ord ; i++) { // Lambda(0)=1 for (i = 1; i < GF.ord ; i++) { // Lambda(0)=1
x = (ui8_t)i; // roll-over x = (ui8_t)i; // roll-over
if (poly_eval(Lambda, x) == 0) { if (poly_eval(sigLam, x) == 0) { // Lambda(x)=0 fuer x in erasures[] moeglich
// error location index // error location index
err_pos[n] = log_a[GF_inv(x)]; err_pos[nerr] = log_a[GF_inv(x)];
// error value; bin-BCH: err_val=1 // error value; bin-BCH: err_val=1
err_val[n] = forney(x, Omega, Lambda); err_val[nerr] = forney(x, Omega, sigLam);
n++; //err_val[nerr] == 0, wenn era_val[pos]=0, d.h. cw[pos] schon korrekt
nerr++;
} }
if (n >= poly_deg(Lambda)) break; if (nerr >= deg_sigLam) break;
} }
if (n < poly_deg(Lambda)) errors = -1; // uncorrectable errors // 2*Errors + Erasure <= 2*RS.t
if (nerr < deg_sigLam) errera = -1; // uncorrectable errors
else { else {
errors = n; errera = nerr;
for (i = 0; i < errors; i++) cw[err_pos[i]] ^= err_val[i]; for (i = 0; i < errera; i++) cw[err_pos[i]] ^= err_val[i];
} }
} }
return errors; return errera;
} }
// Errors <= RS.t
int rs_decode(ui8_t cw[], ui8_t *err_pos, ui8_t *err_val) {
ui8_t tmp[1] = {0};
return rs_decode_ErrEra(cw, 0, tmp, err_pos, err_val);
}
int rs_decode_bch_gf2t2(ui8_t cw[], ui8_t *err_pos, ui8_t *err_val) { int rs_decode_bch_gf2t2(ui8_t cw[], ui8_t *err_pos, ui8_t *err_val) {
// binary 2-error correcting BCH // binary 2-error correcting BCH
@ -795,7 +857,6 @@ int rs_decode_bch_gf2t2(ui8_t cw[], ui8_t *err_pos, ui8_t *err_val) {
L[MAX_DEG+1], L2, L[MAX_DEG+1], L2,
Lambda[MAX_DEG+1], Lambda[MAX_DEG+1],
Omega[MAX_DEG+1]; Omega[MAX_DEG+1];
int i, n, errors = 0; int i, n, errors = 0;
@ -807,7 +868,7 @@ int rs_decode_bch_gf2t2(ui8_t cw[], ui8_t *err_pos, ui8_t *err_val) {
// wenn S(x)=0 , dann poly_divmod(cw, RS.g, d, rem): rem=0 // wenn S(x)=0 , dann poly_divmod(cw, RS.g, d, rem): rem=0
if (errors) { if (errors) {
polyGF_lfsr(RS.t, S, Lambda, Omega); polyGF_lfsr(RS.t, 2*RS.t, S, Lambda, Omega);
gamma = Lambda[0]; gamma = Lambda[0];
if (gamma) { if (gamma) {
for (i = poly_deg(Lambda); i >= 0; i--) Lambda[i] = GF_mul(Lambda[i], GF_inv(gamma)); for (i = poly_deg(Lambda); i >= 0; i--) Lambda[i] = GF_mul(Lambda[i], GF_inv(gamma));