epan/reedsolomon.c

   1 /* reedsolomon.c
   2  *
   3  * Reed-Solomon encoding and decoding,
   4  * by Phil Karn (karn@ka9q.ampr.org) September 1996
   5  * Copyright 1999 Phil Karn, KA9Q
   6  * Separate CCSDS version create Dec 1998, merged into this version May 1999
   7  *
   8  * This file is derived from my generic RS encoder/decoder, which is
   9  * in turn based on the program "new_rs_erasures.c" by Robert
  10  * Morelos-Zaragoza (robert@spectra.eng.hawaii.edu) and Hari Thirumoorthy
  11  * (harit@spectra.eng.hawaii.edu), Aug 1995
  12  *
  13  * Wireshark - Network traffic analyzer
  14  * By Gerald Combs <gerald@wireshark.org>
  15  * Copyright 1998 Gerald Combs
  16  *
  17  * SPDX-License-Identifier: GPL-2.0-or-later
  18  */
  19
  20 #include "config.h"
  21
  22 #define WS_LOG_DOMAIN LOG_DOMAIN_EPAN
  23
  24 #include <stdio.h>
  25 #include "reedsolomon.h"
  26 #include <wsutil/wslog.h>
  27
  28 #ifdef CCSDS
  29 /* CCSDS field generator polynomial: 1+x+x^2+x^7+x^8 */
  30 int Pp[MM+1] = { 1, 1, 1, 0, 0, 0, 0, 1, 1 };
  31
  32 #else /* not CCSDS */
  33 /* MM, KK, B0, PRIM are user-defined in rs.h */
  34
  35 /* Primitive polynomials - see Lin & Costello, Appendix A,
  36  * and  Lee & Messerschmitt, p. 453.
  37  */
  38 #if(MM == 2)/* Admittedly silly */
  39 int Pp[MM+1] = { 1, 1, 1 };
  40
  41 #elif(MM == 3)
  42 /* 1 + x + x^3 */
  43 int Pp[MM+1] = { 1, 1, 0, 1 };
  44
  45 #elif(MM == 4)
  46 /* 1 + x + x^4 */
  47 int Pp[MM+1] = { 1, 1, 0, 0, 1 };
  48
  49 #elif(MM == 5)
  50 /* 1 + x^2 + x^5 */
  51 int Pp[MM+1] = { 1, 0, 1, 0, 0, 1 };
  52
  53 #elif(MM == 6)
  54 /* 1 + x + x^6 */
  55 int Pp[MM+1] = { 1, 1, 0, 0, 0, 0, 1 };
  56
  57 #elif(MM == 7)
  58 /* 1 + x^3 + x^7 */
  59 int Pp[MM+1] = { 1, 0, 0, 1, 0, 0, 0, 1 };
  60
  61 #elif(MM == 8)
  62 /* 1+x^2+x^3+x^4+x^8 */
  63 int Pp[MM+1] = { 1, 0, 1, 1, 1, 0, 0, 0, 1 };
  64
  65 #elif(MM == 9)
  66 /* 1+x^4+x^9 */
  67 int Pp[MM+1] = { 1, 0, 0, 0, 1, 0, 0, 0, 0, 1 };
  68
  69 #elif(MM == 10)
  70 /* 1+x^3+x^10 */
  71 int Pp[MM+1] = { 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1 };
  72
  73 #elif(MM == 11)
  74 /* 1+x^2+x^11 */
  75 int Pp[MM+1] = { 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1 };
  76
  77 #elif(MM == 12)
  78 /* 1+x+x^4+x^6+x^12 */
  79 int Pp[MM+1] = { 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1 };
  80
  81 #elif(MM == 13)
  82 /* 1+x+x^3+x^4+x^13 */
  83 int Pp[MM+1] = { 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1 };
  84
  85 #elif(MM == 14)
  86 /* 1+x+x^6+x^10+x^14 */
  87 int Pp[MM+1] = { 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1 };
  88
  89 #elif(MM == 15)
  90 /* 1+x+x^15 */
  91 int Pp[MM+1] = { 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 };
  92
  93 #elif(MM == 16)
  94 /* 1+x+x^3+x^12+x^16 */
  95 int Pp[MM+1] = { 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1 };
  96
  97 #else
  98 #error "Either CCSDS must be defined, or MM must be set in range 2-16"
  99 #endif
 100
 101 #endif
 102
 103 #ifdef STANDARD_ORDER /* first byte transmitted is index of x**(KK-1) in message poly*/
 104         /* definitions used in the encode routine*/
 105         #define MESSAGE(i) data[KK-(i)-1]
 106         #define REMAINDER(i) bb[NN-KK-(i)-1]
 107         /* definitions used in the decode routine*/
 108         #define RECEIVED(i) data[NN-1-(i)]
 109         #define ERAS_INDEX(i) (NN-1-eras_pos[i])
 110         #define INDEX_TO_POS(i) (NN-1-(i))
 111 #else /* first byte transmitted is index of x**0 in message polynomial*/
 112         /* definitions used in the encode routine*/
 113         #define MESSAGE(i) data[i]
 114         #define REMAINDER(i) bb[i]
 115         /* definitions used in the decode routine*/
 116         #define RECEIVED(i) data[i]
 117         #define ERAS_INDEX(i) eras_pos[i]
 118         #define INDEX_TO_POS(i) i
 119 #endif
 120
 121
 122 /* This defines the type used to store an element of the Galois Field
 123  * used by the code. Make sure this is something larger than a char if
 124  * if anything larger than GF(256) is used.
 125  *
 126  * Note: unsigned char will work up to GF(256) but int seems to run
 127  * faster on the Pentium.
 128  */
 129 typedef int gf;
 130
 131 /* index->polynomial form conversion table */
 132 static gf Alpha_to[NN + 1];
 133
 134 /* Polynomial->index form conversion table */
 135 static gf Index_of[NN + 1];
 136
 137 /* No legal value in index form represents zero, so
 138  * we need a special value for this purpose
 139  */
 140 #define A0      (NN)
 141
 142 /* Generator polynomial g(x) in index form */
 143 static gf Gg[NN - KK + 1];
 144
 145 static int RS_init; /* Initialization flag */
 146
 147 /* Compute x % NN, where NN is 2**MM - 1,
 148  * without a slow divide
 149  */
 150 /* static inline gf*/
 151 static gf
 152 modnn(int x)
 153 {
 154   while (x >= NN) {
 155     x -= NN;
 156     x = (x >> MM) + (x & NN);
 157   }
 158   return x;
 159 }
 160
 161 #define min_(a,b)       ((a) < (b) ? (a) : (b))
 162
 163 #define CLEAR(a,n) {\
 164 int ci;\
 165 for(ci=(n)-1;ci >=0;ci--)\
 166 (a)[ci] = 0;\
 167 }
 168
 169 #define COPY(a,b,n) {\
 170 int ci;\
 171 for(ci=(n)-1;ci >=0;ci--)\
 172 (a)[ci] = (b)[ci];\
 173 }
 174
 175 #define COPYDOWN(a,b,n) {\
 176 int ci;\
 177 for(ci=(n)-1;ci >=0;ci--)\
 178 (a)[ci] = (b)[ci];\
 179 }
 180
 181 static void init_rs(void);
 182
 183 #ifdef CCSDS
 184 /* Conversion lookup tables from conventional alpha to Berlekamp's
 185  * dual-basis representation. Used in the CCSDS version only.
 186  * taltab[] -- convert conventional to dual basis
 187  * tal1tab[] -- convert dual basis to conventional
 188
 189  * Note: the actual RS encoder/decoder works with the conventional basis.
 190  * So data is converted from dual to conventional basis before either
 191  * encoding or decoding and then converted back.
 192  */
 193 static unsigned char taltab[NN+1],tal1tab[NN+1];
 194
 195 static unsigned char tal[] = { 0x8d, 0xef, 0xec, 0x86, 0xfa, 0x99, 0xaf, 0x7b };
 196
 197 /* Generate conversion lookup tables between conventional alpha representation
 198  * (@**7, @**6, ...@**0)
 199  *  and Berlekamp's dual basis representation
 200  * (l0, l1, ...l7)
 201  */
 202 static void
 203 gen_ltab(void)
 204 {
 205   int i,j,k;
 206
 207   for(i=0;i<256;i++){/* For each value of input */
 208     taltab[i] = 0;
 209     for(j=0;j<8;j++) /* for each column of matrix */
 210       for(k=0;k<8;k++){ /* for each row of matrix */
 211         if(i & (1<<k))
 212           taltab[i] ^= tal[7-k] & (1<<j);
 213       }
 214     tal1tab[taltab[i]] = i;
 215   }
 216 }
 217 #endif /* CCSDS */
 218
 219 #if PRIM != 1
 220 static int Ldec;/* Decrement for aux location variable in Chien search */
 221
 222 static void
 223 gen_ldec(void)
 224 {
 225   for(Ldec=1;(Ldec % PRIM) != 0;Ldec+= NN)
 226     ;
 227   Ldec /= PRIM;
 228 }
 229 #else
 230 #define Ldec 1
 231 #endif
 232
 233 /* generate GF(2**m) from the irreducible polynomial p(X) in Pp[0]..Pp[m]
 234    lookup tables:  index->polynomial form   alpha_to[] contains j=alpha**i;
 235                    polynomial form -> index form  index_of[j=alpha**i] = i
 236    alpha=2 is the primitive element of GF(2**m)
 237    HARI's COMMENT: (4/13/94) alpha_to[] can be used as follows:
 238         Let @ represent the primitive element commonly called "alpha" that
 239    is the root of the primitive polynomial p(x). Then in GF(2^m), for any
 240    0 <= i <= 2^m-2,
 241         @^i = a(0) + a(1) @ + a(2) @^2 + ... + a(m-1) @^(m-1)
 242    where the binary vector (a(0),a(1),a(2),...,a(m-1)) is the representation
 243    of the integer "alpha_to[i]" with a(0) being the LSB and a(m-1) the MSB. Thus for
 244    example the polynomial representation of @^5 would be given by the binary
 245    representation of the integer "alpha_to[5]".
 246                    Similarly, index_of[] can be used as follows:
 247         As above, let @ represent the primitive element of GF(2^m) that is
 248    the root of the primitive polynomial p(x). In order to find the power
 249    of @ (alpha) that has the polynomial representation
 250         a(0) + a(1) @ + a(2) @^2 + ... + a(m-1) @^(m-1)
 251    we consider the integer "i" whose binary representation with a(0) being LSB
 252    and a(m-1) MSB is (a(0),a(1),...,a(m-1)) and locate the entry
 253    "index_of[i]". Now, @^index_of[i] is that element whose polynomial
 254     representation is (a(0),a(1),a(2),...,a(m-1)).
 255    NOTE:
 256         The element alpha_to[2^m-1] = 0 always signifying that the
 257    representation of "@^infinity" = 0 is (0,0,0,...,0).
 258         Similarly, the element index_of[0] = A0 always signifying
 259    that the power of alpha which has the polynomial representation
 260    (0,0,...,0) is "infinity".
 261
 262 */
 263
 264 static void
 265 generate_gf(void)
 266 {
 267   register int i, mask;
 268
 269   mask = 1;
 270   Alpha_to[MM] = 0;
 271   for (i = 0; i < MM; i++) {
 272     Alpha_to[i] = mask;
 273     Index_of[Alpha_to[i]] = i;
 274     /* If Pp[i] == 1 then, term @^i occurs in poly-repr of @^MM */
 275     if (Pp[i] != 0)
 276       Alpha_to[MM] ^= mask;     /* Bit-wise EXOR operation */
 277     mask <<= 1; /* single left-shift */
 278   }
 279   Index_of[Alpha_to[MM]] = MM;
 280   /*
 281    * Have obtained poly-repr of @^MM. Poly-repr of @^(i+1) is given by
 282    * poly-repr of @^i shifted left one-bit and accounting for any @^MM
 283    * term that may occur when poly-repr of @^i is shifted.
 284    */
 285   mask >>= 1;
 286   for (i = MM + 1; i < NN; i++) {
 287     if (Alpha_to[i - 1] >= mask)
 288       Alpha_to[i] = Alpha_to[MM] ^ ((Alpha_to[i - 1] ^ mask) << 1);
 289     else
 290       Alpha_to[i] = Alpha_to[i - 1] << 1;
 291     Index_of[Alpha_to[i]] = i;
 292   }
 293   Index_of[0] = A0;
 294   Alpha_to[NN] = 0;
 295 }
 296
 297 /*
 298  * Obtain the generator polynomial of the TT-error correcting, length
 299  * NN=(2**MM -1) Reed Solomon code from the product of (X+@**(B0+i)), i = 0,
 300  * ... ,(2*TT-1)
 301  *
 302  * Examples:
 303  *
 304  * If B0 = 1, TT = 1. deg(g(x)) = 2*TT = 2.
 305  * g(x) = (x+@) (x+@**2)
 306  *
 307  * If B0 = 0, TT = 2. deg(g(x)) = 2*TT = 4.
 308  * g(x) = (x+1) (x+@) (x+@**2) (x+@**3)
 309  */
 310 static void
 311 gen_poly(void)
 312 {
 313   register int i, j;
 314
 315   Gg[0] = 1;
 316   for (i = 0; i < NN - KK; i++) {
 317     Gg[i+1] = 1;
 318     /*
 319      * Below multiply (Gg[0]+Gg[1]*x + ... +Gg[i]x^i) by
 320      * (@**(B0+i)*PRIM + x)
 321      */
 322     for (j = i; j > 0; j--)
 323       if (Gg[j] != 0)
 324         Gg[j] = Gg[j - 1] ^ Alpha_to[modnn((Index_of[Gg[j]]) + (B0 + i) *PRIM)];
 325       else
 326         Gg[j] = Gg[j - 1];
 327     /* Gg[0] can never be zero */
 328     Gg[0] = Alpha_to[modnn(Index_of[Gg[0]] + (B0 + i) * PRIM)];
 329   }
 330   /* convert Gg[] to index form for quicker encoding */
 331   for (i = 0; i <= NN - KK; i++)
 332     Gg[i] = Index_of[Gg[i]];
 333 }
 334
 335
 336 /*
 337  * take the string of symbols in data[i], i=0..(k-1) and encode
 338  * systematically to produce NN-KK parity symbols in bb[0]..bb[NN-KK-1] data[]
 339  * is input and bb[] is output in polynomial form. Encoding is done by using
 340  * a feedback shift register with appropriate connections specified by the
 341  * elements of Gg[], which was generated above. Codeword is   c(X) =
 342  * data(X)*X**(NN-KK)+ b(X)
 343  */
 344
 345 int
 346 encode_rs(dtype data[], dtype bb[])
 347 {
 348   register int i, j;
 349   gf feedback;
 350
 351 #if DEBUG >= 1 && MM != 8
 352   /* Check for illegal input values */
 353   for(i=0;i<KK;i++)
 354     if(MESSAGE(i) > NN)
 355       return -1;
 356 #endif
 357
 358   if(!RS_init)
 359     init_rs();
 360
 361   CLEAR(bb,NN-KK);
 362
 363 #ifdef CCSDS
 364   /* Convert to conventional basis */
 365   for(i=0;i<KK;i++)
 366     MESSAGE(i) = tal1tab[MESSAGE(i)];
 367 #endif
 368
 369   for(i = KK - 1; i >= 0; i--) {
 370     feedback = Index_of[MESSAGE(i) ^ REMAINDER(NN - KK - 1)];
 371     if (feedback != A0) {       /* feedback term is non-zero */
 372       for (j = NN - KK - 1; j > 0; j--)
 373         if (Gg[j] != A0)
 374           REMAINDER(j) = REMAINDER(j - 1) ^ Alpha_to[modnn(Gg[j] + feedback)];
 375         else
 376           REMAINDER(j) = REMAINDER(j - 1);
 377       REMAINDER(0) = Alpha_to[modnn(Gg[0] + feedback)];
 378     } else {    /* feedback term is zero. encoder becomes a
 379                  * single-byte shifter */
 380       for (j = NN - KK - 1; j > 0; j--)
 381         REMAINDER(j) = REMAINDER(j - 1);
 382       REMAINDER(0) = 0;
 383     }
 384   }
 385 #ifdef CCSDS
 386   /* Convert to l-basis */
 387   for(i=0;i<NN;i++)
 388     MESSAGE(i) = taltab[MESSAGE(i)];
 389 #endif
 390
 391   return 0;
 392 }
 393
 394 /*
 395  * Performs ERRORS+ERASURES decoding of RS codes. If decoding is successful,
 396  * writes the codeword into data[] itself. Otherwise data[] is unaltered.
 397  *
 398  * Return number of symbols corrected, or -1 if codeword is illegal
 399  * or uncorrectable. If eras_pos is non-null, the detected error locations
 400  * are written back. NOTE! This array must be at least NN-KK elements long.
 401  *
 402  * First "no_eras" erasures are declared by the calling program. Then, the
 403  * maximum # of errors correctable is t_after_eras = floor((NN-KK-no_eras)/2).
 404  * If the number of channel errors is not greater than "t_after_eras" the
 405  * transmitted codeword will be recovered. Details of algorithm can be found
 406  * in R. Blahut's "Theory ... of Error-Correcting Codes".
 407
 408  * Warning: the eras_pos[] array must not contain duplicate entries; decoder failure
 409  * will result. The decoder *could* check for this condition, but it would involve
 410  * extra time on every decoding operation.
 411  */
 412
 413 int
 414 eras_dec_rs(dtype data[], int eras_pos[], int no_eras)
 415 {
 416   int deg_lambda, el, deg_omega;
 417   int i, j, r,k;
 418   gf u,q,tmp,num1,num2,den,discr_r;
 419   gf lambda[NN-KK + 1], s[NN-KK + 1];   /* Err+Eras Locator poly
 420                                          * and syndrome poly */
 421   gf b[NN-KK + 1], t[NN-KK + 1], omega[NN-KK + 1];
 422   gf root[NN-KK], reg[NN-KK + 1], loc[NN-KK];
 423   int syn_error, count;
 424
 425   if(!RS_init)
 426     init_rs();
 427
 428 #ifdef CCSDS
 429   /* Convert to conventional basis */
 430   for(i=0;i<NN;i++)
 431     RECEIVED(i) = tal1tab[RECEIVED(i)];
 432 #endif
 433
 434 #if DEBUG >= 1 && MM != 8
 435   /* Check for illegal input values */
 436   for(i=0;i<NN;i++)
 437     if(RECEIVED(i) > NN)
 438       return -1;
 439 #endif
 440   /* form the syndromes; i.e., evaluate data(x) at roots of g(x)
 441    * namely @**(B0+i)*PRIM, i = 0, ... ,(NN-KK-1)
 442    */
 443   for(i=1;i<=NN-KK;i++){
 444     s[i] = RECEIVED(0);
 445   }
 446   for(j=1;j<NN;j++){
 447     if(RECEIVED(j) == 0)
 448       continue;
 449     tmp = Index_of[RECEIVED(j)];
 450
 451     /*  s[i] ^= Alpha_to[modnn(tmp + (B0+i-1)*j)]; */
 452     for(i=1;i<=NN-KK;i++)
 453       s[i] ^= Alpha_to[modnn(tmp + (B0+i-1)*PRIM*j)];
 454   }
 455   /* Convert syndromes to index form, checking for nonzero condition */
 456   syn_error = 0;
 457   for(i=1;i<=NN-KK;i++){
 458     syn_error |= s[i];
 459         /*ws_debug("syndrome %d = %x\n",i,s[i]);*/
 460     s[i] = Index_of[s[i]];
 461   }
 462
 463   if (!syn_error) {
 464     /* if syndrome is zero, data[] is a codeword and there are no
 465      * errors to correct. So return data[] unmodified
 466      */
 467     count = 0;
 468     goto finish;
 469   }
 470   CLEAR(&lambda[1],NN-KK);
 471   lambda[0] = 1;
 472
 473   if (no_eras > 0) {
 474     /* Init lambda to be the erasure locator polynomial */
 475     lambda[1] = Alpha_to[modnn(PRIM * ERAS_INDEX(0))];
 476     for (i = 1; i < no_eras; i++) {
 477       u = modnn(PRIM*ERAS_INDEX(i));
 478       for (j = i+1; j > 0; j--) {
 479         tmp = Index_of[lambda[j - 1]];
 480         if(tmp != A0)
 481           lambda[j] ^= Alpha_to[modnn(u + tmp)];
 482       }
 483     }
 484 #if DEBUG >= 1
 485     /* Test code that verifies the erasure locator polynomial just constructed
 486        Needed only for decoder debugging. */
 487
 488     /* find roots of the erasure location polynomial */
 489     for(i=1;i<=no_eras;i++)
 490       reg[i] = Index_of[lambda[i]];
 491     count = 0;
 492     for (i = 1,k=NN-Ldec; i <= NN; i++,k = modnn(NN+k-Ldec)) {
 493       q = 1;
 494       for (j = 1; j <= no_eras; j++)
 495         if (reg[j] != A0) {
 496           reg[j] = modnn(reg[j] + j);
 497           q ^= Alpha_to[reg[j]];
 498         }
 499       if (q != 0)
 500         continue;
 501       /* store root and error location number indices */
 502       root[count] = i;
 503       loc[count] = k;
 504       count++;
 505     }
 506     if (count != no_eras) {
 507       ws_debug("\n lambda(x) is WRONG\n");
 508       count = -1;
 509       goto finish;
 510     }
 511 #if DEBUG >= 2
 512     printf("\n Erasure positions as determined by roots of Eras Loc Poly:\n");
 513     for (i = 0; i < count; i++)
 514       printf("%d ", loc[i]);
 515     printf("\n");
 516 #endif
 517 #endif
 518   }
 519   for(i=0;i<NN-KK+1;i++)
 520     b[i] = Index_of[lambda[i]];
 521
 522   /*
 523    * Begin Berlekamp-Massey algorithm to determine error+erasure
 524    * locator polynomial
 525    */
 526   r = no_eras;
 527   el = no_eras;
 528   while (++r <= NN-KK) {        /* r is the step number */
 529     /* Compute discrepancy at the r-th step in poly-form */
 530     discr_r = 0;
 531     for (i = 0; i < r; i++){
 532       if ((lambda[i] != 0) && (s[r - i] != A0)) {
 533         discr_r ^= Alpha_to[modnn(Index_of[lambda[i]] + s[r - i])];
 534       }
 535     }
 536     discr_r = Index_of[discr_r];        /* Index form */
 537     if (discr_r == A0) {
 538       /* 2 lines below: B(x) <-- x*B(x) */
 539       COPYDOWN(&b[1],b,NN-KK);
 540       b[0] = A0;
 541     } else {
 542       /* 7 lines below: T(x) <-- lambda(x) - discr_r*x*b(x) */
 543       t[0] = lambda[0];
 544       for (i = 0 ; i < NN-KK; i++) {
 545         if(b[i] != A0)
 546           t[i+1] = lambda[i+1] ^ Alpha_to[modnn(discr_r + b[i])];
 547         else
 548           t[i+1] = lambda[i+1];
 549       }
 550       if (2 * el <= r + no_eras - 1) {
 551         el = r + no_eras - el;
 552         /*
 553          * 2 lines below: B(x) <-- inv(discr_r) *
 554          * lambda(x)
 555          */
 556         for (i = 0; i <= NN-KK; i++)
 557           b[i] = (lambda[i] == 0) ? A0 : modnn(Index_of[lambda[i]] - discr_r + NN);
 558       } else {
 559         /* 2 lines below: B(x) <-- x*B(x) */
 560         COPYDOWN(&b[1],b,NN-KK);
 561         b[0] = A0;
 562       }
 563       COPY(lambda,t,NN-KK+1);
 564     }
 565   }
 566
 567   /* Convert lambda to index form and compute deg(lambda(x)) */
 568   deg_lambda = 0;
 569   for(i=0;i<NN-KK+1;i++){
 570     lambda[i] = Index_of[lambda[i]];
 571     if(lambda[i] != A0)
 572       deg_lambda = i;
 573   }
 574   /*
 575    * Find roots of the error+erasure locator polynomial by Chien
 576    * Search
 577    */
 578   COPY(&reg[1],&lambda[1],NN-KK);
 579   count = 0;            /* Number of roots of lambda(x) */
 580   for (i = 1,k=NN-Ldec; i <= NN; i++,k = modnn(NN+k-Ldec)) {
 581     q = 1;
 582     for (j = deg_lambda; j > 0; j--){
 583       if (reg[j] != A0) {
 584         reg[j] = modnn(reg[j] + j);
 585         q ^= Alpha_to[reg[j]];
 586       }
 587     }
 588     if (q != 0)
 589       continue;
 590     /* store root (index-form) and error location number */
 591     root[count] = i;
 592     loc[count] = k;
 593     /* If we've already found max possible roots,
 594      * abort the search to save time
 595      */
 596     if(++count == deg_lambda)
 597       break;
 598   }
 599   if (deg_lambda != count) {
 600     /*
 601      * deg(lambda) unequal to number of roots => uncorrectable
 602      * error detected
 603      */
 604     count = -1;
 605     goto finish;
 606   }
 607   /*
 608    * Compute err+eras evaluator poly omega(x) = s(x)*lambda(x) (modulo
 609    * x**(NN-KK)). in index form. Also find deg(omega).
 610    */
 611   deg_omega = 0;
 612   for (i = 0; i < NN-KK;i++){
 613     tmp = 0;
 614     j = (deg_lambda < i) ? deg_lambda : i;
 615     for(;j >= 0; j--){
 616       if ((s[i + 1 - j] != A0) && (lambda[j] != A0))
 617         tmp ^= Alpha_to[modnn(s[i + 1 - j] + lambda[j])];
 618     }
 619     if(tmp != 0)
 620       deg_omega = i;
 621     omega[i] = Index_of[tmp];
 622   }
 623   omega[NN-KK] = A0;
 624
 625   /*
 626    * Compute error values in poly-form. num1 = omega(inv(X(l))), num2 =
 627    * inv(X(l))**(B0-1) and den = lambda_pr(inv(X(l))) all in poly-form
 628    */
 629   for (j = count-1; j >=0; j--) {
 630     num1 = 0;
 631     for (i = deg_omega; i >= 0; i--) {
 632       if (omega[i] != A0)
 633         num1  ^= Alpha_to[modnn(omega[i] + i * root[j])];
 634     }
 635     num2 = Alpha_to[modnn(root[j] * (B0 - 1) + NN)];
 636     den = 0;
 637
 638     /* lambda[i+1] for i even is the formal derivative lambda_pr of lambda[i] */
 639     for (i = min_(deg_lambda,NN-KK-1) & ~1; i >= 0; i -=2) {
 640       if(lambda[i+1] != A0)
 641         den ^= Alpha_to[modnn(lambda[i+1] + i * root[j])];
 642     }
 643     if (den == 0) {
 644 #if DEBUG >= 1
 645       ws_debug("\n ERROR: denominator = 0\n");
 646 #endif
 647       /* Convert to dual- basis */
 648       count = -1;
 649       goto finish;
 650     }
 651     /* Apply error to data */
 652     if (num1 != 0) {
 653       RECEIVED(loc[j]) ^= Alpha_to[modnn(Index_of[num1] + Index_of[num2] + NN - Index_of[den])];
 654     }
 655   }
 656  finish:
 657 #ifdef CCSDS
 658   /* Convert to dual- basis */
 659   for(i=0;i<NN;i++)
 660     RECEIVED(i) = taltab[RECEIVED(i)];
 661 #endif
 662   if(eras_pos != NULL){
 663     for(i=0;i<count;i++){
 664       if(eras_pos!= NULL)
 665         eras_pos[i] = INDEX_TO_POS(loc[i]);
 666     }
 667   }
 668   return count;
 669 }
 670 /* Encoder/decoder initialization - call this first! */
 671 static void
 672 init_rs(void)
 673 {
 674   generate_gf();
 675   gen_poly();
 676 #ifdef CCSDS
 677   gen_ltab();
 678 #endif
 679 #if PRIM != 1
 680   gen_ldec();
 681 #endif
 682   RS_init = 1;
 683 }
 684
 685 /*
 686  * Editor modelines  -  https://www.wireshark.org/tools/modelines.html
 687  *
 688  * Local Variables:
 689  * c-basic-offset: 2
 690  * tab-width: 8
 691  * indent-tabs-mode: nil
 692  * End:
 693  *
 694  * ex: set shiftwidth=2 tabstop=8 expandtab:
 695  * :indentSize=2:tabSize=8:noTabs=true:
 696  */