I have been trying to implement a Reed Solomon encoder to produce code words in systematic form with the message followed by the check symbols. For comparison, I have referenced this whitepaper:http://www.bbc.co.uk/rd/pubs/whp/whp031.shtml which is very instructive with hand calculations and LFSR description.
For the example given in this whitepaper the primitive polynomial of the RS(15,11) code is X^4 X 1 and the message is 1,2,3,4,5,6,7,8,9,10,11 and the resulting check symbols are 3,3,12,12.
I can verify this by hand. I have also used this code https://en.wikiversity.org/wiki/Reed–Solomon_codes_for_coders referenced by the Wikipedia article on Reed Solomon to verify that the check symbols are 3,3,12,12.
However, when I use the Python galois package and this code I get different results. The check symbols are 11,10,14,6.
import galois
rs = galois.ReedSolomon(15,11, primitive_poly = 19)
GF = rs.field;
## Encode the message
m = GF([1,2,3,4,5,6,7,8,9,10,11])
c = rs.encode(m)
This result agrees with MATLAB.
Are there two different algorithms to encode a systematic codeword in reed solomon that yield different check symbols? If not, which method is correct?
CodePudding user response：
The issue is the generator polynomials are different.
Using generator polynomial (x-1)(x-2)(x-4)(x-8), the encode produces 3,3,12,12.
Using generator polynomial (x-2)(x-4)(x-8)(x-3), the encode produces 11,10,14,6.
Both generator polynomials are "correct". The generator that starts with (x-2) is termed to be a "narrow sense code".