#include "rs.h"
extern int pp[mm+1]; /* specify irreducible polynomial coeffts */
extern int alpha_to[nn+1], index_of[nn+1], gg[nn-kk+1] ;
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< 256; 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(int recd[nn], int data[kk], int bb[nn-kk])
/* 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 ;
} ;
} ;
for (i=0; i<nn-kk; i++)
recd[i] = bb[i] ;
for (i=0; i<kk; i++)
recd[i+nn-kk] = data[i] ;
}
void decode_rs(int recd[nn], int dataout[kk])
/* 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
{
u++ ;
if (d[u]==-1)
{
l[u+1] = l[u] ;
for (i=0; i<=l[u]; i++)
{
elp[u+1][i] = elp[u][i] ;
elp[u][i] = index_of[elp[u][i]] ;
}
}
else
/* search for words with greatest u_lu[q] for which d[q]!=0 */
{
q = u-1 ;
while ((d[q]==-1) && (q>0))
q-- ;
/* have found first non-zero d[q] */
if (q>0)
{
j=q ;
do
{
j-- ;
if ((d[j]!=-1) && (u_lu[q]<u_lu[j]))
q = j ;
}while (j>0) ;
} ;
/* have now found q such that d[u]!=0 and u_lu[q] is maximum */
/* store degree of new elp polynomial */
if (l[u]>l[q]+u-q)
l[u+1] = l[u] ;
else
l[u+1] = l[q]+u-q ;
/* form new elp(x) */
for (i=0; i<nn-kk; i++)
elp[u+1][i] = 0 ;
for (i=0; i<=l[q]; i++)
if (elp[q][i]!=-1)
elp[u+1][i+u-q] = alpha_to[(d[u]+nn-d[q]+elp[q][i])%nn] ;
for (i=0; i<=l[u]; i++)
{
elp[u+1][i] ^= elp[u][i] ;
elp[u][i] = index_of[elp[u][i]] ; /*convert old elp value to index*/
}
}
u_lu[u+1] = u-l[u+1] ;
/* form (u+1)th discrepancy */
if (u<nn-kk) /* no discrepancy computed on last iteration */
{
if (s[u+1]!=-1)
d[u+1] = alpha_to[s[u+1]] ;
else
d[u+1] = 0 ;
for (i=1; i<=l[u+1]; i++)
if ((s[u+1-i]!=-1) && (elp[u+1][i]!=0))
d[u+1] ^= alpha_to[(s[u+1-i]+index_of[elp[u+1][i]])%nn] ;
d[u+1] = index_of[d[u+1]] ; /* put d[u+1] into index form */
}
} while ((u<nn-kk) && (l[u+1]<=tt)) ;
u++ ;
if (l[u]<=tt) /* can correct error */
{
/* put elp into index form */
for (i=0; i<=l[u]; i++)
elp[u][i] = index_of[elp[u][i]] ;
/* find roots of the error location polynomial */
for (i=1; i<=l[u]; i++)
reg[i] = elp[u][i] ;
count = 0 ;
for (i=1; i<=nn; i++)
{
q = 1 ;
for (j=1; j<=l[u]; j++)
if (reg[j]!=-1)
{
reg[j] = (reg[j]+j)%nn ;
q ^= alpha_to[reg[j]] ;
} ;
if (!q) /* store root and error location number indices */
{
root[count] = i;
loc[count] = nn-i ;
count++ ;
};
} ;
if (count==l[u]) /* no. roots = degree of elp hence <= tt errors */
{
/* form polynomial z(x) */
for (i=1; i<=l[u]; i++) /* Z[0] = 1 always - do not need */
{
if ((s[i]!=-1) && (elp[u][i]!=-1))
z[i] = alpha_to[s[i]] ^ alpha_to[elp[u][i]] ;
else if ((s[i]!=-1) && (elp[u][i]==-1))
z[i] = alpha_to[s[i]] ;
else if ((s[i]==-1) && (elp[u][i]!=-1))
z[i] = alpha_to[elp[u][i]] ;
else
z[i] = 0 ;
for (j=1; j<i; j++)
if ((s[j]!=-1) && (elp[u][i-j]!=-1))
z
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