In this website I consider two main classes of error-correcting codes which I refer to as "Algebraic Codes" and "Low-Density Parity-Check (LDPC) Codes".
Algebraic codes are codes where the decoder, except for burst-trapping decoders, does computations on elements in a finite field to determine what message was most-likely to have been sent.
LDPC codes are codes where the decoder, for most applications, computes probabilities that are conventional numbers and uses inference to correct errors.