Finite Field Arithmetic and Reed-Solomon Coding

This article is great---but would anyone like to recommend a good reference to implementing Reed-Solomon (and other FEC schemes?) that <i>includes</i> decoding?

This just got my brain jazzed. Any other good resources or uses of Finite Fields that would interest a programmer?

One minor nitpick: the set of numbers mod 256 form a field. Powers of primes are also fields.<p>Edit: it seems I conflated two different constructions of a field of order 256. Apologies. It's been a few semesters since abstract algebra &#62;.&#60;

A small summary:<p>1. Z/pZ is a field if and only if p is prime.<p>2. for every p prime and n&#62;=1 there exists a unique (up to isomorphism) field, called Galois field (see any book of algebra for the proof).<p>3. you can build a field of p^n elements for every p and n&#62;1, using polynomials over Z/pZ mod an irreducible polynomial of degree n, e.g. (see link) you can build F_{2^8} as polynomials with coefficients in Z_2 (i.e. bits) mod x^8 + x^4 + x^3 + x + 1. If you chose another irreducible polynomial, e.g. x^8 + x^4 + x^3 + x^2 + 1, then you get another representation of a field of 256 elements, but "structurally" they are the same (this should "explain" the expression "up to isomorphism")

Oh, BTW, if you want to implement finite fields arithmetic (at least for bigger fields), don't follow this link -and don't use go-, but see the code of any crypto library, e.g. openssl (yet there are faster implementations).