/* * ECC algorithm for M-systems disk on chip. We use the excellent Reed * Solmon code of Phil Karn (karn@ka9q.ampr.org) available under the * GNU GPL License. The rest is simply to convert the disk on chip * syndrom into a standard syndom. * * Author: Fabrice Bellard (fabrice.bellard@netgem.com) * Copyright (C) 2000 Netgem S.A. * * $Id: docecc.c,v 1.4 2001/10/02 15:05:13 dwmw2 Exp $ * * This program is free software; you can redistribute it and/or modify * it under the terms of the GNU General Public License as published by * the Free Software Foundation; either version 2 of the License, or * (at your option) any later version. * * This program is distributed in the hope that it will be useful, * but WITHOUT ANY WARRANTY; without even the implied warranty of * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the * GNU General Public License for more details. * * You should have received a copy of the GNU General Public License * along with this program; if not, write to the Free Software * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */ #include <config.h> #include <common.h> #include <malloc.h> #undef ECC_DEBUG #undef PSYCHO_DEBUG #include <linux/mtd/doc2000.h> /* need to undef it (from asm/termbits.h) */ #undef B0 #define MM 10 /* Symbol size in bits */ #define KK (1023-4) /* Number of data symbols per block */ #define B0 510 /* First root of generator polynomial, alpha form */ #define PRIM 1 /* power of alpha used to generate roots of generator poly */ #define NN ((1 << MM) - 1) typedef unsigned short dtype; /* 1+x^3+x^10 */ static const int Pp[MM+1] = { 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1 }; /* This defines the type used to store an element of the Galois Field * used by the code. Make sure this is something larger than a char if * if anything larger than GF(256) is used. * * Note: unsigned char will work up to GF(256) but int seems to run * faster on the Pentium. */ typedef int gf; /* No legal value in index form represents zero, so * we need a special value for this purpose */ #define A0 (NN) /* 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 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];\ } #define Ldec 1 /* generate GF(2**m) from the irreducible polynomial p(X) in Pp[0]..Pp[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(dtype Alpha_to[NN + 1], dtype Index_of[NN + 1]) { 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; } /* * Performs ERRORS+ERASURES decoding of RS codes. bb[] is the content * of the feedback shift register after having processed the data and * the ECC. * * Return number of symbols corrected, or -1 if codeword is illegal * or uncorrectable. If eras_pos is non-null, the detected error locations * are written back. NOTE! This array must be at least NN-KK elements long. * The corrected data are written in eras_val[]. They must be xor with the data * to retrieve the correct data : data[erase_pos[i]] ^= erase_val[i] . * * First "no_eras" erasures are declared by the calling program. Then, the * maximum # of errors correctable is t_after_eras = floor((NN-KK-no_eras)/2). * If the number of channel errors is not greater than "t_after_eras" the * transmitted codeword will be recovered. Details of algorithm can be found * in R. Blahut's "Theory ... of Error-Correcting Codes". * Warning: the eras_pos[] array must not contain duplicate entries; decoder failure * will result. The decoder *could* check for this condition, but it would involve * extra time on every decoding operation. * */ static int eras_dec_rs(dtype Alpha_to[NN + 1], dtype Index_of[NN + 1], gf bb[NN - KK + 1], gf eras_val[NN-KK], int eras_pos[NN-KK], int no_eras) { int deg_lambda, el, deg_omega; int i, j, r,k; 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], loc[NN-KK]; int syn_error, count; syn_error = 0; for(i=0;i<NN-KK;i++) syn_error |= bb[i]; if (!syn_error) { /* if remainder is zero, data[] is a codeword and there are no * errors to correct. So return data[] unmodified */ count = 0; goto finish; } for(i=1;i<=NN-KK;i++){ s[i] = bb[0]; } for(j=1;j<NN-KK;j++){ if(bb[j] == 0) continue; tmp = Index_of[bb[j]]; for(i=1;i<=NN-KK;i++) s[i] ^= Alpha_to[modnn(tmp + (B0+i-1)*PRIM*j)]; } /* undo the feedback register implicit multiplication and convert syndromes to index form */ for(i=1;i<=NN-KK;i++) { tmp = Index_of[s[i]]; if (tmp != A0) tmp = modnn(tmp + 2 * KK * (B0+i-1)*PRIM); s[i] = tmp; } CLEAR(&lambda[1],NN-KK); lambda[0] = 1; if (no_eras > 0) { /* Init lambda to be the erasure locator polynomial */ lambda[1] = Alpha_to[modnn(PRIM * eras_pos[0])]; for (i = 1; i < no_eras; i++) { u = modnn(PRIM*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)]; } } #ifdef ECC_DEBUG /* Test code that verifies the erasure locator polynomial just constructed Needed only for decoder debugging. */ /* find roots of the erasure location polynomial */ for(i=1;i<=no_eras;i++) reg[i] = Index_of[lambda[i]]; count = 0; for (i = 1,k=NN-Ldec; i <= NN; i++,k = modnn(NN+k-Ldec)) { q = 1; for (j = 1; j <= no_eras; j++) if (reg[j] != A0) { reg[j] = modnn(reg[j] + j); q ^= Alpha_to[reg[j]]; } if (q != 0) continue; /* store root and error location number indices */ root[count] = i; loc[count] = k; count++; } if (count != no_eras) { printf("\n lambda(x) is WRONG\n"); count = -1; goto finish; } #ifdef PSYCHO_DEBUG printf("\n Erasure positions as determined by roots of Eras Loc Poly:\n"); for (i = 0; i < count; i++) printf("%d ", loc[i]); printf("\n"); #endif #endif } 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,k=NN-Ldec; i <= NN; i++,k = modnn(NN+k-Ldec)) { 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 != 0) continue; /* store root (index-form) and error location number */ root[count] = i; loc[count] = k; /* If we've already found max possible roots, * abort the search to save time */ if(++count == deg_lambda) break; } if (deg_lambda != count) { /* * deg(lambda) unequal to number of roots => uncorrectable * error detected */ count = -1; goto finish; } /* * 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 ECC_DEBUG printf("\n ERROR: denominator = 0\n"); #endif /* Convert to dual- basis */ count = -1; goto finish; } /* Apply error to data */ if (num1 != 0) { eras_val[j] = Alpha_to[modnn(Index_of[num1] + Index_of[num2] + NN - Index_of[den])]; } else { eras_val[j] = 0; } } finish: for(i=0;i<count;i++) eras_pos[i] = loc[i]; return count; } /***************************************************************************/ /* The DOC specific code begins here */ #define SECTOR_SIZE 512 /* The sector bytes are packed into NB_DATA MM bits words */ #define NB_DATA (((SECTOR_SIZE + 1) * 8 + 6) / MM) /* * Correct the errors in 'sector[]' by using 'ecc1[]' which is the * content of the feedback shift register applyied to the sector and * the ECC. Return the number of errors corrected (and correct them in * sector), or -1 if error */ int doc_decode_ecc(unsigned char sector[SECTOR_SIZE], unsigned char ecc1[6]) { int parity, i, nb_errors; gf bb[NN - KK + 1]; gf error_val[NN-KK]; int error_pos[NN-KK], pos, bitpos, index, val; dtype *Alpha_to, *Index_of; /* init log and exp tables here to save memory. However, it is slower */ Alpha_to = malloc((NN + 1) * sizeof(dtype)); if (!Alpha_to) return -1; Index_of = malloc((NN + 1) * sizeof(dtype)); if (!Index_of) { free(Alpha_to); return -1; } generate_gf(Alpha_to, Index_of); parity = ecc1[1]; bb[0] = (ecc1[4] & 0xff) | ((ecc1[5] & 0x03) << 8); bb[1] = ((ecc1[5] & 0xfc) >> 2) | ((ecc1[2] & 0x0f) << 6); bb[2] = ((ecc1[2] & 0xf0) >> 4) | ((ecc1[3] & 0x3f) << 4); bb[3] = ((ecc1[3] & 0xc0) >> 6) | ((ecc1[0] & 0xff) << 2); nb_errors = eras_dec_rs(Alpha_to, Index_of, bb, error_val, error_pos, 0); if (nb_errors <= 0) goto the_end; /* correct the errors */ for(i=0;i<nb_errors;i++) { pos = error_pos[i]; if (pos >= NB_DATA && pos < KK) { nb_errors = -1; goto the_end; } if (pos < NB_DATA) { /* extract bit position (MSB first) */ pos = 10 * (NB_DATA - 1 - pos) - 6; /* now correct the following 10 bits. At most two bytes can be modified since pos is even */ index = (pos >> 3) ^ 1; bitpos = pos & 7; if ((index >= 0 && index < SECTOR_SIZE) || index == (SECTOR_SIZE + 1)) { val = error_val[i] >> (2 + bitpos); parity ^= val; if (index < SECTOR_SIZE) sector[index] ^= val; } index = ((pos >> 3) + 1) ^ 1; bitpos = (bitpos + 10) & 7; if (bitpos == 0) bitpos = 8; if ((index >= 0 && index < SECTOR_SIZE) || index == (SECTOR_SIZE + 1)) { val = error_val[i] << (8 - bitpos); parity ^= val; if (index < SECTOR_SIZE) sector[index] ^= val; } } } /* use parity to test extra errors */ if ((parity & 0xff) != 0) nb_errors = -1; the_end: free(Alpha_to); free(Index_of); return nb_errors; }