Algebraic Codes

Algebraic error-correcting codes use finite field arithmetic from modern algebra for encoding and decoding.

Algebraic codes can correct all combinations of t or fewer errors.  When t > 1, more redundancy is required than is absolutely necessary, but the extra redundancy can be used to reduce the probability of miscorrection if decoders are designed a certain way.

Algebraic codes can correct all combination of t or fewer errors if the minimum Hamming distance (or just minimum distance) is > 2t.  The Hamming distance between two codewords is the number of positions in which they differ.

Algebraic and probabilistic codes have a number of things in common.  They are both linear block codes and use a generator matrix, G,  for encoding and a parity-check matrix, H,  for computing parity checks (or syndrome).

Understanding algebraic codes requires knowledge of finite fields from modern algebra.