2
* Reed-Solomon coding and decoding
3
* Phil Karn (karn@ka9q.ampr.org) September 1996
4
* Separate CCSDS version create Dec 1998, merged into this version May 1999
6
* This file is derived from my generic RS encoder/decoder, which is
7
* in turn based on the program "new_rs_erasures.c" by Robert
8
* Morelos-Zaragoza (robert@spectra.eng.hawaii.edu) and Hari Thirumoorthy
9
* (harit@spectra.eng.hawaii.edu), Aug 1995
11
* Copyright 1999 Phil Karn, KA9Q
12
* May be used under the terms of the GNU public license
15
#include "reedsolomon.h"
18
/* CCSDS field generator polynomial: 1+x+x^2+x^7+x^8 */
19
int Pp[MM+1] = { 1, 1, 1, 0, 0, 0, 0, 1, 1 };
22
/* MM, KK, B0, PRIM are user-defined in rs.h */
24
/* Primitive polynomials - see Lin & Costello, Appendix A,
25
* and Lee & Messerschmitt, p. 453.
27
#if(MM == 2)/* Admittedly silly */
28
int Pp[MM+1] = { 1, 1, 1 };
32
int Pp[MM+1] = { 1, 1, 0, 1 };
36
int Pp[MM+1] = { 1, 1, 0, 0, 1 };
40
int Pp[MM+1] = { 1, 0, 1, 0, 0, 1 };
44
int Pp[MM+1] = { 1, 1, 0, 0, 0, 0, 1 };
48
int Pp[MM+1] = { 1, 0, 0, 1, 0, 0, 0, 1 };
51
/* 1+x^2+x^3+x^4+x^8 */
52
int Pp[MM+1] = { 1, 0, 1, 1, 1, 0, 0, 0, 1 };
56
int Pp[MM+1] = { 1, 0, 0, 0, 1, 0, 0, 0, 0, 1 };
60
int Pp[MM+1] = { 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1 };
64
int Pp[MM+1] = { 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1 };
67
/* 1+x+x^4+x^6+x^12 */
68
int Pp[MM+1] = { 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1 };
71
/* 1+x+x^3+x^4+x^13 */
72
int Pp[MM+1] = { 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1 };
75
/* 1+x+x^6+x^10+x^14 */
76
int Pp[MM+1] = { 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1 };
80
int Pp[MM+1] = { 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 };
83
/* 1+x+x^3+x^12+x^16 */
84
int Pp[MM+1] = { 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1 };
87
#error "Either CCSDS must be defined, or MM must be set in range 2-16"
92
#ifdef STANDARD_ORDER /* first byte transmitted is index of x**(KK-1) in message poly*/
93
/* definitions used in the encode routine*/
94
#define MESSAGE(i) data[KK-(i)-1]
95
#define REMAINDER(i) bb[NN-KK-(i)-1]
96
/* definitions used in the decode routine*/
97
#define RECEIVED(i) data[NN-1-(i)]
98
#define ERAS_INDEX(i) (NN-1-eras_pos[i])
99
#define INDEX_TO_POS(i) (NN-1-(i))
100
#else /* first byte transmitted is index of x**0 in message polynomial*/
101
/* definitions used in the encode routine*/
102
#define MESSAGE(i) data[i]
103
#define REMAINDER(i) bb[i]
104
/* definitions used in the decode routine*/
105
#define RECEIVED(i) data[i]
106
#define ERAS_INDEX(i) eras_pos[i]
107
#define INDEX_TO_POS(i) i
111
/* This defines the type used to store an element of the Galois Field
112
* used by the code. Make sure this is something larger than a char if
113
* if anything larger than GF(256) is used.
115
* Note: unsigned char will work up to GF(256) but int seems to run
116
* faster on the Pentium.
120
/* index->polynomial form conversion table */
121
static gf Alpha_to[NN + 1];
123
/* Polynomial->index form conversion table */
124
static gf Index_of[NN + 1];
126
/* No legal value in index form represents zero, so
127
* we need a special value for this purpose
131
/* Generator polynomial g(x) in index form */
132
static gf Gg[NN - KK + 1];
134
static int RS_init; /* Initialization flag */
136
/* Compute x % NN, where NN is 2**MM - 1,
137
* without a slow divide
139
/* static inline gf*/
145
x = (x >> MM) + (x & NN);
150
#define min_(a,b) ((a) < (b) ? (a) : (b))
152
#define CLEAR(a,n) {\
154
for(ci=(n)-1;ci >=0;ci--)\
158
#define COPY(a,b,n) {\
160
for(ci=(n)-1;ci >=0;ci--)\
164
#define COPYDOWN(a,b,n) {\
166
for(ci=(n)-1;ci >=0;ci--)\
170
static void init_rs(void);
173
/* Conversion lookup tables from conventional alpha to Berlekamp's
174
* dual-basis representation. Used in the CCSDS version only.
175
* taltab[] -- convert conventional to dual basis
176
* tal1tab[] -- convert dual basis to conventional
178
* Note: the actual RS encoder/decoder works with the conventional basis.
179
* So data is converted from dual to conventional basis before either
180
* encoding or decoding and then converted back.
182
static unsigned char taltab[NN+1],tal1tab[NN+1];
184
static unsigned char tal[] = { 0x8d, 0xef, 0xec, 0x86, 0xfa, 0x99, 0xaf, 0x7b };
186
/* Generate conversion lookup tables between conventional alpha representation
187
* (@**7, @**6, ...@**0)
188
* and Berlekamp's dual basis representation
196
for(i=0;i<256;i++){/* For each value of input */
198
for(j=0;j<8;j++) /* for each column of matrix */
199
for(k=0;k<8;k++){ /* for each row of matrix */
201
taltab[i] ^= tal[7-k] & (1<<j);
203
tal1tab[taltab[i]] = i;
209
static int Ldec;/* Decrement for aux location variable in Chien search */
214
for(Ldec=1;(Ldec % PRIM) != 0;Ldec+= NN)
222
/* generate GF(2**m) from the irreducible polynomial p(X) in Pp[0]..Pp[m]
223
lookup tables: index->polynomial form alpha_to[] contains j=alpha**i;
224
polynomial form -> index form index_of[j=alpha**i] = i
225
alpha=2 is the primitive element of GF(2**m)
226
HARI's COMMENT: (4/13/94) alpha_to[] can be used as follows:
227
Let @ represent the primitive element commonly called "alpha" that
228
is the root of the primitive polynomial p(x). Then in GF(2^m), for any
230
@^i = a(0) + a(1) @ + a(2) @^2 + ... + a(m-1) @^(m-1)
231
where the binary vector (a(0),a(1),a(2),...,a(m-1)) is the representation
232
of the integer "alpha_to[i]" with a(0) being the LSB and a(m-1) the MSB. Thus for
233
example the polynomial representation of @^5 would be given by the binary
234
representation of the integer "alpha_to[5]".
235
Similarily, index_of[] can be used as follows:
236
As above, let @ represent the primitive element of GF(2^m) that is
237
the root of the primitive polynomial p(x). In order to find the power
238
of @ (alpha) that has the polynomial representation
239
a(0) + a(1) @ + a(2) @^2 + ... + a(m-1) @^(m-1)
240
we consider the integer "i" whose binary representation with a(0) being LSB
241
and a(m-1) MSB is (a(0),a(1),...,a(m-1)) and locate the entry
242
"index_of[i]". Now, @^index_of[i] is that element whose polynomial
243
representation is (a(0),a(1),a(2),...,a(m-1)).
245
The element alpha_to[2^m-1] = 0 always signifying that the
246
representation of "@^infinity" = 0 is (0,0,0,...,0).
247
Similarily, the element index_of[0] = A0 always signifying
248
that the power of alpha which has the polynomial representation
249
(0,0,...,0) is "infinity".
256
register int i, mask;
260
for (i = 0; i < MM; i++) {
262
Index_of[Alpha_to[i]] = i;
263
/* If Pp[i] == 1 then, term @^i occurs in poly-repr of @^MM */
265
Alpha_to[MM] ^= mask; /* Bit-wise EXOR operation */
266
mask <<= 1; /* single left-shift */
268
Index_of[Alpha_to[MM]] = MM;
270
* Have obtained poly-repr of @^MM. Poly-repr of @^(i+1) is given by
271
* poly-repr of @^i shifted left one-bit and accounting for any @^MM
272
* term that may occur when poly-repr of @^i is shifted.
275
for (i = MM + 1; i < NN; i++) {
276
if (Alpha_to[i - 1] >= mask)
277
Alpha_to[i] = Alpha_to[MM] ^ ((Alpha_to[i - 1] ^ mask) << 1);
279
Alpha_to[i] = Alpha_to[i - 1] << 1;
280
Index_of[Alpha_to[i]] = i;
287
* Obtain the generator polynomial of the TT-error correcting, length
288
* NN=(2**MM -1) Reed Solomon code from the product of (X+@**(B0+i)), i = 0,
293
* If B0 = 1, TT = 1. deg(g(x)) = 2*TT = 2.
294
* g(x) = (x+@) (x+@**2)
296
* If B0 = 0, TT = 2. deg(g(x)) = 2*TT = 4.
297
* g(x) = (x+1) (x+@) (x+@**2) (x+@**3)
305
for (i = 0; i < NN - KK; i++) {
308
* Below multiply (Gg[0]+Gg[1]*x + ... +Gg[i]x^i) by
309
* (@**(B0+i)*PRIM + x)
311
for (j = i; j > 0; j--)
313
Gg[j] = Gg[j - 1] ^ Alpha_to[modnn((Index_of[Gg[j]]) + (B0 + i) *PRIM)];
316
/* Gg[0] can never be zero */
317
Gg[0] = Alpha_to[modnn(Index_of[Gg[0]] + (B0 + i) * PRIM)];
319
/* convert Gg[] to index form for quicker encoding */
320
for (i = 0; i <= NN - KK; i++)
321
Gg[i] = Index_of[Gg[i]];
326
* take the string of symbols in data[i], i=0..(k-1) and encode
327
* systematically to produce NN-KK parity symbols in bb[0]..bb[NN-KK-1] data[]
328
* is input and bb[] is output in polynomial form. Encoding is done by using
329
* a feedback shift register with appropriate connections specified by the
330
* elements of Gg[], which was generated above. Codeword is c(X) =
331
* data(X)*X**(NN-KK)+ b(X)
335
encode_rs(dtype data[KK], dtype bb[NN-KK])
340
#if DEBUG >= 1 && MM != 8
341
/* Check for illegal input values */
353
/* Convert to conventional basis */
355
MESSAGE(i) = tal1tab[MESSAGE(i)];
358
for(i = KK - 1; i >= 0; i--) {
359
feedback = Index_of[MESSAGE(i) ^ REMAINDER(NN - KK - 1)];
360
if (feedback != A0) { /* feedback term is non-zero */
361
for (j = NN - KK - 1; j > 0; j--)
363
REMAINDER(j) = REMAINDER(j - 1) ^ Alpha_to[modnn(Gg[j] + feedback)];
365
REMAINDER(j) = REMAINDER(j - 1);
366
REMAINDER(0) = Alpha_to[modnn(Gg[0] + feedback)];
367
} else { /* feedback term is zero. encoder becomes a
368
* single-byte shifter */
369
for (j = NN - KK - 1; j > 0; j--)
370
REMAINDER(j) = REMAINDER(j - 1);
375
/* Convert to l-basis */
377
MESSAGE(i) = taltab[MESSAGE(i)];
384
* Performs ERRORS+ERASURES decoding of RS codes. If decoding is successful,
385
* writes the codeword into data[] itself. Otherwise data[] is unaltered.
387
* Return number of symbols corrected, or -1 if codeword is illegal
388
* or uncorrectable. If eras_pos is non-null, the detected error locations
389
* are written back. NOTE! This array must be at least NN-KK elements long.
391
* First "no_eras" erasures are declared by the calling program. Then, the
392
* maximum # of errors correctable is t_after_eras = floor((NN-KK-no_eras)/2).
393
* If the number of channel errors is not greater than "t_after_eras" the
394
* transmitted codeword will be recovered. Details of algorithm can be found
395
* in R. Blahut's "Theory ... of Error-Correcting Codes".
397
* Warning: the eras_pos[] array must not contain duplicate entries; decoder failure
398
* will result. The decoder *could* check for this condition, but it would involve
399
* extra time on every decoding operation.
403
eras_dec_rs(dtype data[NN], int eras_pos[NN-KK], int no_eras)
405
int deg_lambda, el, deg_omega;
407
gf u,q,tmp,num1,num2,den,discr_r;
408
gf lambda[NN-KK + 1], s[NN-KK + 1]; /* Err+Eras Locator poly
409
* and syndrome poly */
410
gf b[NN-KK + 1], t[NN-KK + 1], omega[NN-KK + 1];
411
gf root[NN-KK], reg[NN-KK + 1], loc[NN-KK];
412
int syn_error, count;
418
/* Convert to conventional basis */
420
RECEIVED(i) = tal1tab[RECEIVED(i)];
423
#if DEBUG >= 1 && MM != 8
424
/* Check for illegal input values */
429
/* form the syndromes; i.e., evaluate data(x) at roots of g(x)
430
* namely @**(B0+i)*PRIM, i = 0, ... ,(NN-KK-1)
432
for(i=1;i<=NN-KK;i++){
438
tmp = Index_of[RECEIVED(j)];
440
/* s[i] ^= Alpha_to[modnn(tmp + (B0+i-1)*j)]; */
441
for(i=1;i<=NN-KK;i++)
442
s[i] ^= Alpha_to[modnn(tmp + (B0+i-1)*PRIM*j)];
444
/* Convert syndromes to index form, checking for nonzero condition */
446
for(i=1;i<=NN-KK;i++){
448
/*printf("syndrome %d = %x\n",i,s[i]);*/
449
s[i] = Index_of[s[i]];
453
/* if syndrome is zero, data[] is a codeword and there are no
454
* errors to correct. So return data[] unmodified
459
CLEAR(&lambda[1],NN-KK);
463
/* Init lambda to be the erasure locator polynomial */
464
lambda[1] = Alpha_to[modnn(PRIM * ERAS_INDEX(0))];
465
for (i = 1; i < no_eras; i++) {
466
u = modnn(PRIM*ERAS_INDEX(i));
467
for (j = i+1; j > 0; j--) {
468
tmp = Index_of[lambda[j - 1]];
470
lambda[j] ^= Alpha_to[modnn(u + tmp)];
474
/* Test code that verifies the erasure locator polynomial just constructed
475
Needed only for decoder debugging. */
477
/* find roots of the erasure location polynomial */
478
for(i=1;i<=no_eras;i++)
479
reg[i] = Index_of[lambda[i]];
481
for (i = 1,k=NN-Ldec; i <= NN; i++,k = modnn(NN+k-Ldec)) {
483
for (j = 1; j <= no_eras; j++)
485
reg[j] = modnn(reg[j] + j);
486
q ^= Alpha_to[reg[j]];
490
/* store root and error location number indices */
495
if (count != no_eras) {
496
printf("\n lambda(x) is WRONG\n");
501
printf("\n Erasure positions as determined by roots of Eras Loc Poly:\n");
502
for (i = 0; i < count; i++)
503
printf("%d ", loc[i]);
508
for(i=0;i<NN-KK+1;i++)
509
b[i] = Index_of[lambda[i]];
512
* Begin Berlekamp-Massey algorithm to determine error+erasure
517
while (++r <= NN-KK) { /* r is the step number */
518
/* Compute discrepancy at the r-th step in poly-form */
520
for (i = 0; i < r; i++){
521
if ((lambda[i] != 0) && (s[r - i] != A0)) {
522
discr_r ^= Alpha_to[modnn(Index_of[lambda[i]] + s[r - i])];
525
discr_r = Index_of[discr_r]; /* Index form */
527
/* 2 lines below: B(x) <-- x*B(x) */
528
COPYDOWN(&b[1],b,NN-KK);
531
/* 7 lines below: T(x) <-- lambda(x) - discr_r*x*b(x) */
533
for (i = 0 ; i < NN-KK; i++) {
535
t[i+1] = lambda[i+1] ^ Alpha_to[modnn(discr_r + b[i])];
537
t[i+1] = lambda[i+1];
539
if (2 * el <= r + no_eras - 1) {
540
el = r + no_eras - el;
542
* 2 lines below: B(x) <-- inv(discr_r) *
545
for (i = 0; i <= NN-KK; i++)
546
b[i] = (lambda[i] == 0) ? A0 : modnn(Index_of[lambda[i]] - discr_r + NN);
548
/* 2 lines below: B(x) <-- x*B(x) */
549
COPYDOWN(&b[1],b,NN-KK);
552
COPY(lambda,t,NN-KK+1);
556
/* Convert lambda to index form and compute deg(lambda(x)) */
558
for(i=0;i<NN-KK+1;i++){
559
lambda[i] = Index_of[lambda[i]];
564
* Find roots of the error+erasure locator polynomial by Chien
567
COPY(®[1],&lambda[1],NN-KK);
568
count = 0; /* Number of roots of lambda(x) */
569
for (i = 1,k=NN-Ldec; i <= NN; i++,k = modnn(NN+k-Ldec)) {
571
for (j = deg_lambda; j > 0; j--){
573
reg[j] = modnn(reg[j] + j);
574
q ^= Alpha_to[reg[j]];
579
/* store root (index-form) and error location number */
582
/* If we've already found max possible roots,
583
* abort the search to save time
585
if(++count == deg_lambda)
588
if (deg_lambda != count) {
590
* deg(lambda) unequal to number of roots => uncorrectable
597
* Compute err+eras evaluator poly omega(x) = s(x)*lambda(x) (modulo
598
* x**(NN-KK)). in index form. Also find deg(omega).
601
for (i = 0; i < NN-KK;i++){
603
j = (deg_lambda < i) ? deg_lambda : i;
605
if ((s[i + 1 - j] != A0) && (lambda[j] != A0))
606
tmp ^= Alpha_to[modnn(s[i + 1 - j] + lambda[j])];
610
omega[i] = Index_of[tmp];
615
* Compute error values in poly-form. num1 = omega(inv(X(l))), num2 =
616
* inv(X(l))**(B0-1) and den = lambda_pr(inv(X(l))) all in poly-form
618
for (j = count-1; j >=0; j--) {
620
for (i = deg_omega; i >= 0; i--) {
622
num1 ^= Alpha_to[modnn(omega[i] + i * root[j])];
624
num2 = Alpha_to[modnn(root[j] * (B0 - 1) + NN)];
627
/* lambda[i+1] for i even is the formal derivative lambda_pr of lambda[i] */
628
for (i = min_(deg_lambda,NN-KK-1) & ~1; i >= 0; i -=2) {
629
if(lambda[i+1] != A0)
630
den ^= Alpha_to[modnn(lambda[i+1] + i * root[j])];
634
printf("\n ERROR: denominator = 0\n");
636
/* Convert to dual- basis */
640
/* Apply error to data */
642
RECEIVED(loc[j]) ^= Alpha_to[modnn(Index_of[num1] + Index_of[num2] + NN - Index_of[den])];
647
/* Convert to dual- basis */
649
RECEIVED(i) = taltab[RECEIVED(i)];
651
if(eras_pos != NULL){
652
for(i=0;i<count;i++){
654
eras_pos[i] = INDEX_TO_POS(loc[i]);
659
/* Encoder/decoder initialization - call this first! */
added
removed
epan/reedsolomon.c
