Algebraic codes use finite field mathematics from modern algebra.  See the subpages for examples of finite fields and their use in RS Codes and RS RAID.

Probabilistic codes use normal arithmetic to compute conditional probabilities.  Operations are usually performed on log likelihood ratios (LLRs).  LLRs are signed numbers with the property that if the LLR is positive the associated bit is most-likely a 1 and if the LLR is negative, the associated bit is most-likely a 0 or vice versa which makes it extremely simple to convert a string of LLRs into a string of most-likely bits.