2 * Copyright (c) 2001-2003, David Janssens
3 * Copyright (c) 2002-2003, Yannick Verschueren
4 * Copyright (c) 2003-2005, Francois Devaux and Antonin Descampe
5 * Copyright (c) 2005, Herv� Drolon, FreeImage Team
6 * Copyright (c) 2002-2005, Communications and remote sensing Laboratory, Universite catholique de Louvain, Belgium
7 * Copyright (c) 2005-2006, Dept. of Electronic and Information Engineering, Universita' degli Studi di Perugia, Italy
10 * Redistribution and use in source and binary forms, with or without
11 * modification, are permitted provided that the following conditions
13 * 1. Redistributions of source code must retain the above copyright
14 * notice, this list of conditions and the following disclaimer.
15 * 2. Redistributions in binary form must reproduce the above copyright
16 * notice, this list of conditions and the following disclaimer in the
17 * documentation and/or other materials provided with the distribution.
19 * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS `AS IS'
20 * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
21 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
22 * ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
23 * LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
24 * CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
25 * SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
26 * INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
27 * CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
28 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
29 * POSSIBILITY OF SUCH DAMAGE.
36 @brief Functions used to compute the Reed-Solomon parity and check of byte arrays
41 * Reed-Solomon coding and decoding
42 * Phil Karn (karn@ka9q.ampr.org) September 1996
44 * This file is derived from the program "new_rs_erasures.c" by Robert
45 * Morelos-Zaragoza (robert@spectra.eng.hawaii.edu) and Hari Thirumoorthy
46 * (harit@spectra.eng.hawaii.edu), Aug 1995
48 * I've made changes to improve performance, clean up the code and make it
49 * easier to follow. Data is now passed to the encoding and decoding functions
50 * through arguments rather than in global arrays. The decode function returns
51 * the number of corrected symbols, or -1 if the word is uncorrectable.
53 * This code supports a symbol size from 2 bits up to 16 bits,
54 * implying a block size of 3 2-bit symbols (6 bits) up to 65535
55 * 16-bit symbols (1,048,560 bits). The code parameters are set in rs.h.
57 * Note that if symbols larger than 8 bits are used, the type of each
58 * data array element switches from unsigned char to unsigned int. The
59 * caller must ensure that elements larger than the symbol range are
60 * not passed to the encoder or decoder.
67 /* This defines the type used to store an element of the Galois Field
68 * used by the code. Make sure this is something larger than a char if
69 * if anything larger than GF(256) is used.
71 * Note: unsigned char will work up to GF(256) but int seems to run
72 * faster on the Pentium.
76 /* Primitive polynomials - see Lin & Costello, Appendix A,
77 * and Lee & Messerschmitt, p. 453.
79 #if(MM == 2)/* Admittedly silly */
80 int Pp[MM+1] = { 1, 1, 1 };
84 int Pp[MM+1] = { 1, 1, 0, 1 };
88 int Pp[MM+1] = { 1, 1, 0, 0, 1 };
92 int Pp[MM+1] = { 1, 0, 1, 0, 0, 1 };
96 int Pp[MM+1] = { 1, 1, 0, 0, 0, 0, 1 };
100 int Pp[MM+1] = { 1, 0, 0, 1, 0, 0, 0, 1 };
103 /* 1+x^2+x^3+x^4+x^8 */
104 int Pp[MM+1] = { 1, 0, 1, 1, 1, 0, 0, 0, 1 };
108 int Pp[MM+1] = { 1, 0, 0, 0, 1, 0, 0, 0, 0, 1 };
112 int Pp[MM+1] = { 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1 };
116 int Pp[MM+1] = { 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1 };
119 /* 1+x+x^4+x^6+x^12 */
120 int Pp[MM+1] = { 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1 };
123 /* 1+x+x^3+x^4+x^13 */
124 int Pp[MM+1] = { 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1 };
127 /* 1+x+x^6+x^10+x^14 */
128 int Pp[MM+1] = { 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1 };
132 int Pp[MM+1] = { 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 };
135 /* 1+x+x^3+x^12+x^16 */
136 int Pp[MM+1] = { 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1 };
139 #error "MM must be in range 2-16"
142 /* Alpha exponent for the first root of the generator polynomial */
143 #define B0 0 /* Different from the default 1 */
145 /* index->polynomial form conversion table */
148 /* Polynomial->index form conversion table */
151 /* No legal value in index form represents zero, so
152 * we need a special value for this purpose
156 /* Generator polynomial g(x)
157 * Degree of g(x) = 2*TT
158 * has roots @**B0, @**(B0+1), ... ,@^(B0+2*TT-1)
160 /*gf Gg[NN - KK + 1];*/
163 /* Compute x % NN, where NN is 2**MM - 1,
164 * without a slow divide
171 x = (x >> MM) + (x & NN);
176 /*#define min(a,b) ((a) < (b) ? (a) : (b))*/
178 #define CLEAR(a,n) {\
180 for(ci=(n)-1;ci >=0;ci--)\
184 #define COPY(a,b,n) {\
186 for(ci=(n)-1;ci >=0;ci--)\
189 #define COPYDOWN(a,b,n) {\
191 for(ci=(n)-1;ci >=0;ci--)\
199 printf("KK must be less than 2**MM - 1\n");
207 /* generate GF(2**m) from the irreducible polynomial p(X) in p[0]..p[m]
208 lookup tables: index->polynomial form alpha_to[] contains j=alpha**i;
209 polynomial form -> index form index_of[j=alpha**i] = i
210 alpha=2 is the primitive element of GF(2**m)
211 HARI's COMMENT: (4/13/94) alpha_to[] can be used as follows:
212 Let @ represent the primitive element commonly called "alpha" that
213 is the root of the primitive polynomial p(x). Then in GF(2^m), for any
215 @^i = a(0) + a(1) @ + a(2) @^2 + ... + a(m-1) @^(m-1)
216 where the binary vector (a(0),a(1),a(2),...,a(m-1)) is the representation
217 of the integer "alpha_to[i]" with a(0) being the LSB and a(m-1) the MSB. Thus for
218 example the polynomial representation of @^5 would be given by the binary
219 representation of the integer "alpha_to[5]".
220 Similarily, index_of[] can be used as follows:
221 As above, let @ represent the primitive element of GF(2^m) that is
222 the root of the primitive polynomial p(x). In order to find the power
223 of @ (alpha) that has the polynomial representation
224 a(0) + a(1) @ + a(2) @^2 + ... + a(m-1) @^(m-1)
225 we consider the integer "i" whose binary representation with a(0) being LSB
226 and a(m-1) MSB is (a(0),a(1),...,a(m-1)) and locate the entry
227 "index_of[i]". Now, @^index_of[i] is that element whose polynomial
228 representation is (a(0),a(1),a(2),...,a(m-1)).
230 The element alpha_to[2^m-1] = 0 always signifying that the
231 representation of "@^infinity" = 0 is (0,0,0,...,0).
232 Similarily, the element index_of[0] = A0 always signifying
233 that the power of alpha which has the polynomial representation
234 (0,0,...,0) is "infinity".
241 register int i, mask;
245 for (i = 0; i < MM; i++) {
247 Index_of[Alpha_to[i]] = i;
248 /* If Pp[i] == 1 then, term @^i occurs in poly-repr of @^MM */
250 Alpha_to[MM] ^= mask; /* Bit-wise EXOR operation */
251 mask <<= 1; /* single left-shift */
253 Index_of[Alpha_to[MM]] = MM;
255 * Have obtained poly-repr of @^MM. Poly-repr of @^(i+1) is given by
256 * poly-repr of @^i shifted left one-bit and accounting for any @^MM
257 * term that may occur when poly-repr of @^i is shifted.
260 for (i = MM + 1; i < NN; i++) {
261 if (Alpha_to[i - 1] >= mask)
262 Alpha_to[i] = Alpha_to[MM] ^ ((Alpha_to[i - 1] ^ mask) << 1);
264 Alpha_to[i] = Alpha_to[i - 1] << 1;
265 Index_of[Alpha_to[i]] = i;
273 * Obtain the generator polynomial of the TT-error correcting, length
274 * NN=(2**MM -1) Reed Solomon code from the product of (X+@**(B0+i)), i = 0,
279 * If B0 = 1, TT = 1. deg(g(x)) = 2*TT = 2.
280 * g(x) = (x+@) (x+@**2)
282 * If B0 = 0, TT = 2. deg(g(x)) = 2*TT = 4.
283 * g(x) = (x+1) (x+@) (x+@**2) (x+@**3)
290 Gg[0] = Alpha_to[B0];
291 Gg[1] = 1; /* g(x) = (X+@**B0) initially */
292 for (i = 2; i <= NN - KK; i++) {
295 * Below multiply (Gg[0]+Gg[1]*x + ... +Gg[i]x^i) by
298 for (j = i - 1; j > 0; j--)
300 Gg[j] = Gg[j - 1] ^ Alpha_to[modnn((Index_of[Gg[j]]) + B0 + i - 1)];
303 /* Gg[0] can never be zero */
304 Gg[0] = Alpha_to[modnn((Index_of[Gg[0]]) + B0 + i - 1)];
306 /* convert Gg[] to index form for quicker encoding */
307 for (i = 0; i <= NN - KK; i++)
308 Gg[i] = Index_of[Gg[i]];
313 * take the string of symbols in data[i], i=0..(k-1) and encode
314 * systematically to produce NN-KK parity symbols in bb[0]..bb[NN-KK-1] data[]
315 * is input and bb[] is output in polynomial form. Encoding is done by using
316 * a feedback shift register with appropriate connections specified by the
317 * elements of Gg[], which was generated above. Codeword is c(X) =
318 * data(X)*X**(NN-KK)+ b(X)
321 encode_rs(dtype *data, dtype *bb)
327 for (i = KK - 1; i >= 0; i--) {
330 return -1; /* Illegal symbol */
332 feedback = Index_of[data[i] ^ bb[NN - KK - 1]];
333 if (feedback != A0) { /* feedback term is non-zero */
334 for (j = NN - KK - 1; j > 0; j--)
336 bb[j] = bb[j - 1] ^ Alpha_to[modnn(Gg[j] + feedback)];
339 bb[0] = Alpha_to[modnn(Gg[0] + feedback)];
340 } else { /* feedback term is zero. encoder becomes a
341 * single-byte shifter */
342 for (j = NN - KK - 1; j > 0; j--)
351 * Performs ERRORS+ERASURES decoding of RS codes. If decoding is successful,
352 * writes the codeword into data[] itself. Otherwise data[] is unaltered.
354 * Return number of symbols corrected, or -1 if codeword is illegal
357 * First "no_eras" erasures are declared by the calling program. Then, the
358 * maximum # of errors correctable is t_after_eras = floor((NN-KK-no_eras)/2).
359 * If the number of channel errors is not greater than "t_after_eras" the
360 * transmitted codeword will be recovered. Details of algorithm can be found
361 * in R. Blahut's "Theory ... of Error-Correcting Codes".
364 eras_dec_rs(dtype *data, int *eras_pos, int no_eras)
366 int deg_lambda, el, deg_omega;
368 gf u,q,tmp,num1,num2,den,discr_r;
370 /* Err+Eras Locator poly and syndrome poly */
371 /*gf lambda[NN-KK + 1], s[NN-KK + 1];
372 gf b[NN-KK + 1], t[NN-KK + 1], omega[NN-KK + 1];
373 gf root[NN-KK], reg[NN-KK + 1], loc[NN-KK];*/
374 gf lambda[NN + 1], s[NN + 1];
375 gf b[NN + 1], t[NN + 1], omega[NN + 1];
376 gf root[NN], reg[NN + 1], loc[NN];
377 int syn_error, count;
379 /* data[] is in polynomial form, copy and convert to index form */
380 for (i = NN-1; i >= 0; i--){
383 return -1; /* Illegal symbol */
385 recd[i] = Index_of[data[i]];
387 /* first form the syndromes; i.e., evaluate recd(x) at roots of g(x)
388 * namely @**(B0+i), i = 0, ... ,(NN-KK-1)
391 for (i = 1; i <= NN-KK; i++) {
393 for (j = 0; j < NN; j++)
394 if (recd[j] != A0) /* recd[j] in index form */
395 tmp ^= Alpha_to[modnn(recd[j] + (B0+i-1)*j)];
396 syn_error |= tmp; /* set flag if non-zero syndrome =>
398 /* store syndrome in index form */
399 s[i] = Index_of[tmp];
403 * if syndrome is zero, data[] is a codeword and there are no
404 * errors to correct. So return data[] unmodified
408 CLEAR(&lambda[1],NN-KK);
411 /* Init lambda to be the erasure locator polynomial */
412 lambda[1] = Alpha_to[eras_pos[0]];
413 for (i = 1; i < no_eras; i++) {
415 for (j = i+1; j > 0; j--) {
416 tmp = Index_of[lambda[j - 1]];
418 lambda[j] ^= Alpha_to[modnn(u + tmp)];
422 /* find roots of the erasure location polynomial */
423 for(i=1;i<=no_eras;i++)
424 reg[i] = Index_of[lambda[i]];
426 for (i = 1; i <= NN; i++) {
428 for (j = 1; j <= no_eras; j++)
430 reg[j] = modnn(reg[j] + j);
431 q ^= Alpha_to[reg[j]];
434 /* store root and error location
442 if (count != no_eras) {
443 printf("\n lambda(x) is WRONG\n");
447 printf("\n Erasure positions as determined by roots of Eras Loc Poly:\n");
448 for (i = 0; i < count; i++)
449 printf("%d ", loc[i]);
454 for(i=0;i<NN-KK+1;i++)
455 b[i] = Index_of[lambda[i]];
458 * Begin Berlekamp-Massey algorithm to determine error+erasure
463 while (++r <= NN-KK) { /* r is the step number */
464 /* Compute discrepancy at the r-th step in poly-form */
466 for (i = 0; i < r; i++){
467 if ((lambda[i] != 0) && (s[r - i] != A0)) {
468 discr_r ^= Alpha_to[modnn(Index_of[lambda[i]] + s[r - i])];
471 discr_r = Index_of[discr_r]; /* Index form */
473 /* 2 lines below: B(x) <-- x*B(x) */
474 COPYDOWN(&b[1],b,NN-KK);
477 /* 7 lines below: T(x) <-- lambda(x) - discr_r*x*b(x) */
479 for (i = 0 ; i < NN-KK; i++) {
481 t[i+1] = lambda[i+1] ^ Alpha_to[modnn(discr_r + b[i])];
483 t[i+1] = lambda[i+1];
485 if (2 * el <= r + no_eras - 1) {
486 el = r + no_eras - el;
488 * 2 lines below: B(x) <-- inv(discr_r) *
491 for (i = 0; i <= NN-KK; i++)
492 b[i] = (lambda[i] == 0) ? A0 : modnn(Index_of[lambda[i]] - discr_r + NN);
494 /* 2 lines below: B(x) <-- x*B(x) */
495 COPYDOWN(&b[1],b,NN-KK);
498 COPY(lambda,t,NN-KK+1);
502 /* Convert lambda to index form and compute deg(lambda(x)) */
504 for(i=0;i<NN-KK+1;i++){
505 lambda[i] = Index_of[lambda[i]];
510 * Find roots of the error+erasure locator polynomial. By Chien
513 COPY(®[1],&lambda[1],NN-KK);
514 count = 0; /* Number of roots of lambda(x) */
515 for (i = 1; i <= NN; i++) {
517 for (j = deg_lambda; j > 0; j--)
519 reg[j] = modnn(reg[j] + j);
520 q ^= Alpha_to[reg[j]];
523 /* store root (index-form) and error location number */
531 printf("\n Final error positions:\t");
532 for (i = 0; i < count; i++)
533 printf("%d ", loc[i]);
536 if (deg_lambda != count) {
538 * deg(lambda) unequal to number of roots => uncorrectable
544 * Compute err+eras evaluator poly omega(x) = s(x)*lambda(x) (modulo
545 * x**(NN-KK)). in index form. Also find deg(omega).
548 for (i = 0; i < NN-KK;i++){
550 j = (deg_lambda < i) ? deg_lambda : i;
552 if ((s[i + 1 - j] != A0) && (lambda[j] != A0))
553 tmp ^= Alpha_to[modnn(s[i + 1 - j] + lambda[j])];
557 omega[i] = Index_of[tmp];
562 * Compute error values in poly-form. num1 = omega(inv(X(l))), num2 =
563 * inv(X(l))**(B0-1) and den = lambda_pr(inv(X(l))) all in poly-form
565 for (j = count-1; j >=0; j--) {
567 for (i = deg_omega; i >= 0; i--) {
569 num1 ^= Alpha_to[modnn(omega[i] + i * root[j])];
571 num2 = Alpha_to[modnn(root[j] * (B0 - 1) + NN)];
574 /* lambda[i+1] for i even is the formal derivative lambda_pr of lambda[i] */
575 for (i = min(deg_lambda,NN-KK-1) & ~1; i >= 0; i -=2) {
576 if(lambda[i+1] != A0)
577 den ^= Alpha_to[modnn(lambda[i+1] + i * root[j])];
581 printf("\n ERROR: denominator = 0\n");
585 /* Apply error to data */
587 data[loc[j]] ^= Alpha_to[modnn(Index_of[num1] + Index_of[num2] + NN - Index_of[den])];
594 #endif /* USE_JPWL */