Reed Solomon decode with c differnet from '1'

[mhostetter]

Discussed in #214

^{Originally posted by miritom December 28, 2021}
Hi @mhostetter,

Thanks for maintaining this library. I started using it recently and it's very useful.
One thing I noticed is the Reed Solomon decode utility requires c=1 to work properly.
I tried the same example from the documentation, just added "c=2" to the constructor and it failed.
Is this a known limitiation? I saw that there are tests for the "encode" method with different c values, but all "decode" tests use the default, although I dindn't observe any comment on this in the documentation.

Example

In [1]: import galois                                                                                                    

In [2]: rs = galois.ReedSolomon(15, 9, c=2); rs                                                                          
Out[2]: <Reed-Solomon Code: [15, 9, 7] over GF(2^4)>

In [3]: GF = rs.field                                                                                                    

In [4]: m = GF.Random(rs.k); m                                                                                           
Out[4]: GF([10,  6,  3,  8, 11,  1,  1, 13,  1], order=2^4)

In [5]: c = rs.encode(m); c                                                                                              
Out[5]: 
GF([10,  6,  3,  8, 11,  1,  1, 13,  1,  2,  8,  3, 10, 15,  5],
   order=2^4)

In [6]: c[0] += GF(13); c                                                                                                   
Out[6]: 
GF([ 7,  6,  3,  8, 11,  1,  1, 13,  1,  2,  8,  3, 10, 15,  5],
   order=2^4)

# The first symbol is correctly erroneously
In [10]: dec_m = rs.decode(c); dec_m                                                                                     
Out[10]: GF([ 8,  6,  3,  8, 11,  1,  1, 13,  1], order=2^4)

In [11]: m                                                                                                               
Out[11]: GF([10,  6,  3,  8, 11,  1,  1, 13,  1], order=2^4)

[mhostetter]

The issue was

was used in the Forney Algorithm instead of

.

Reference: https://en.wikipedia.org/wiki/Forney_algorithm