linear codes

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linear codes In coding theory, codes whose encoding and decoding operations may be expressed in terms of linear operations. The term is usually applied to certain error-correcting codes in which the encoding operation involves a generator matrix and the decoding operation involves a parity-check matrix. Linear codes are, therefore, also called parity-check codes. A particular linear code forms a commutative group that has the zero codeword as its identity.

In the case of linear (n, k) block codes, the generator matrix is k × n and the parity-check matrix is (n k) × n; the elements of both matrices are elements of the base field (this being error-correcting codes for binary codes). See also convolutional code.