Reed solomon library with coefficent support?

Question

I have to decode some reed solomon codes. I tried to use open source libraries for this but I'm not sure how to apply them to my specific problem.

For example this lib: https://github.com/zxing/zxing/blob/master/core/src/main/java/com/google/zxing/common/reedsolomon/GenericGF.java

Seems to support a lot of codes with a generator function like this:

x^12 + x^6 + x^5 + x^3 + 1
x^10 + x^3 + 1

But I need to support a function with coefficients like this:

2x^10 + 4x^3 + 8

Is my assumption correct that this is not supported by the library? Does anyone know a library that supports this?

Answer 1

Reed-Solomon is formally defined using symbols over any finite field. A popular choice is GF(2^b) for some bit-length b; for example, if the symbols are bytes, you would use GF(2^8). GF(2^8) is implemented using polynomials with coefficients over GF(2), modulo a generator. zxing's GenericGF only works for GF(2^b) fields, in which the polynomial coefficients are all either 0 or 1. Your coefficients are not binary, which suggests that you might be operating over GF(p^b) for some other prime p; in this case, zxing will probably not work for you.

You could try a more generic implementation from a computer algebra system; for example, SageMath provides generalized Reed-Solomon codes: https://doc.sagemath.org/html/en/reference/coding/sage/coding/grs_code.html

Side-note: confusingly, there's another definition of "generator polynomial" in Reed-Solomon: in the BCH scheme, the generator is a polynomial whose coefficients are field elements (e.g. GF(p^b)) and whose value is typically g(x) = (x - a)(x - a^2)(x - a^3)... for some primitive element a of the field. I presume this is not the generator you're referring to, as the provided polynomial is not of the right form.