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. |
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