/*             rs.c        *//* This program is an encoder/decoder for Reed-Solomon codes. Encoding is in   systematic form, decoding via the Berlekamp iterative algorithm.   In the present form , the constants mm, nn, tt, and kk=nn-2tt must be   specified  (the double letters are used simply to avoid clashes with   other n,k,t used in other programs into which this was incorporated!)   Also, the irreducible polynomial used to generate GF(2**mm) must also be   entered -- these can be found in Lin and Costello, and also Clark and Cain.   The representation of the elements of GF(2**m) is either in index form,   where the number is the power of the primitive element alpha, which is   convenient for multiplication (add the powers modulo 2**m-1) or in   polynomial form, where the bits represent the coefficients of the   polynomial representation of the number, which is the most convenient form   for addition.  The two forms are swapped between via lookup tables.   This leads to fairly messy looking expressions, but unfortunately, there   is no easy alternative when working with Galois arithmetic.   The code is not written in the most elegant way, but to the best   of my knowledge, (no absolute guarantees!), it works.   However, when including it into a simulation program, you may want to do   some conversion of global variables (used here because I am lazy!) to   local variables where appropriate, and passing parameters (eg array   addresses) to the functions  may be a sensible move to reduce the number   of global variables and thus decrease the chance of a bug being introduced.   This program does not handle erasures at present, but should not be hard   to adapt to do this, as it is just an adjustment to the Berlekamp-Massey   algorithm. It also does not attempt to decode past the BCH bound -- see   Blahut "Theory and practice of error control codes" for how to do this.              Simon Rockliff, University of Adelaide   21/9/89   26/6/91 Slight modifications to remove a compiler dependent bug which hadn't           previously surfaced. A few extra comments added for clarity.           Appears to all work fine, ready for posting to net!                  Notice                 --------   This program may be freely modified and/or given to whoever wants it.   A condition of such distribution is that the author's contribution be   acknowledged by his name being left in the comments heading the program,   however no responsibility is accepted for any financial or other loss which   may result from some unforseen errors or malfunctioning of the program   during use.                                 Simon Rockliff, 26th June 1991*/#include <math.h>#include <stdio.h>#define mm  4           /* RS code over GF(2**4) - change to suit */#define nn  15          /* nn=2**mm -1   length of codeword */#define tt  3           /* number of errors that can be corrected */#define kk  9           /* kk = nn-2*tt  */int pp [mm+1] = { 1, 1, 0, 0, 1} ; /* specify irreducible polynomial coeffts */int alpha_to [nn+1], index_of [nn+1], gg [nn-kk+1] ;int recd [nn], data [kk], bb [nn-kk] ;void generate_gf()/* generate GF(2**mm) from the irreducible polynomial p(X) in pp[0]..pp[mm]   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**mm)*/ {   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]!=0)       alpha_to[mm] ^= mask ;     mask <<= 1 ;   }  index_of[alpha_to[mm]] = mm ;  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] = -1 ; }void gen_poly()/* Obtain the generator polynomial of the tt-error correcting, length  nn=(2**mm -1) Reed Solomon code  from the product of (X+alpha**i), i=1..2*tt*/ {   register int i,j ;   gg[0] = 2 ;    /* primitive element alpha = 2  for GF(2**mm)  */   gg[1] = 1 ;    /* g(x) = (X+alpha) initially */   for (i=2; i<=nn-kk; i++)    { gg[i] = 1 ;      for (j=i-1; j>0; j--)        if (gg[j] != 0)  gg[j] = gg[j-1]^ alpha_to[(index_of[gg[j]]+i)%nn] ;        else gg[j] = gg[j-1] ;      gg[0] = alpha_to[(index_of[gg[0]]+i)%nn] ;     /* gg[0] can never be zero */    }   /* convert gg[] to index form for quicker encoding */   for (i=0; i<=nn-kk; i++)  gg[i] = index_of[gg[i]] ; }void encode_rs()/* take the string of symbols in data[i], i=0..(k-1) and encode systematically   to produce 2*tt parity symbols in bb[0]..bb[2*tt-1]   data[] is input and bb[] is output in polynomial form.   Encoding is done by using a feedback shift register with appropriate   connections specified by the elements of gg[], which was generated above.   Codeword is   c(X) = data(X)*X**(nn-kk)+ b(X)          */ {   register int i,j ;   int feedback ;   for (i=0; i<nn-kk; i++)   bb[i] = 0 ;   for (i=kk-1; i>=0; i--)    {  feedback = index_of[data[i]^bb[nn-kk-1]] ;       if (feedback != -1)        { for (j=nn-kk-1; j>0; j--)            if (gg[j] != -1)              bb[j] = bb[j-1]^alpha_to[(gg[j]+feedback)%nn] ;            else              bb[j] = bb[j-1] ;          bb[0] = alpha_to[(gg[0]+feedback)%nn] ;        }       else        { for (j=nn-kk-1; j>0; j--)            bb[j] = bb[j-1] ;          bb[0] = 0 ;        } ;    } ; } ;void decode_rs()/* assume we have received bits grouped into mm-bit symbols in recd[i],   i=0..(nn-1),  and recd[i] is index form (ie as powers of alpha).   We first compute the 2*tt syndromes by substituting alpha**i into rec(X) and   evaluating, storing the syndromes in s[i], i=1..2tt (leave s[0] zero) .   Then we use the Berlekamp iteration to find the error location polynomial   elp[i].   If the degree of the elp is >tt, we cannot correct all the errors   and hence just put out the information symbols uncorrected. If the degree of   elp is <=tt, we substitute alpha**i , i=1..n into the elp to get the roots,   hence the inverse roots, the error location numbers. If the number of errors   located does not equal the degree of the elp, we have more than tt errors   and cannot correct them.  Otherwise, we then solve for the error value at   the error location and correct the error.  The procedure is that found in   Lin and Costello. For the cases where the number of errors is known to be too   large to correct, the information symbols as received are output (the   advantage of systematic encoding is that hopefully some of the information   symbols will be okay and that if we are in luck, the errors are in the   parity part of the transmitted codeword).  Of course, these insoluble cases   can be returned as error flags to the calling routine if desired.   */ {   register int i,j,u,q ;   int elp[nn-kk+2][nn-kk], d[nn-kk+2], l[nn-kk+2], u_lu[nn-kk+2], s[nn-kk+1] ;   int count=0, syn_error=0, root[tt], loc[tt], z[tt+1], err[nn], reg[tt+1] ;/* first form the syndromes */   for (i=1; i<=nn-kk; i++)    { s[i] = 0 ;      for (j=0; j<nn; j++)        if (recd[j]!=-1)          s[i] ^= alpha_to[(recd[j]+i*j)%nn] ;      /* recd[j] in index form *//* convert syndrome from polynomial form to index form  */      if (s[i]!=0)  syn_error=1 ;        /* set flag if non-zero syndrome => error */      s[i] = index_of[s[i]] ;    } ;   if (syn_error)       /* if errors, try and correct */    {/* compute the error location polynomial via the Berlekamp iterative algorithm,   following the terminology of Lin and Costello :   d[u] is the 'mu'th   discrepancy, where u='mu'+1 and 'mu' (the Greek letter!) is the step number   ranging from -1 to 2*tt (see L&C),  l[u] is the   degree of the elp at that step, and u_l[u] is the difference between the   step number and the degree of the elp.*//* initialise table entries */      d[0] = 0 ;           /* index form */      d[1] = s[1] ;        /* index form */      elp[0][0] = 0 ;      /* index form */      elp[1][0] = 1 ;      /* polynomial form */      for (i=1; i<nn-kk; i++)        { elp[0][i] = -1 ;   /* index form */          elp[1][i] = 0 ;   /* polynomial form */        }      l[0] = 0 ;      l[1] = 0 ;      u_lu[0] = -1 ;      u_lu[1] = 0 ;      u = 0 ;      do      {