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Reed–Muller codes are linear block codes that are locally testable, locally decodable, and list decodable. These properties make them particularly useful in the design of probabilistically checkable proofs . Traditional Reed–Muller codes are binary codes, which means that messages and codewords are binary strings.
An irreducible code is a cyclic code in which the code, as an ideal is irreducible, i.e. is minimal in , so that its check polynomial is an irreducible polynomial. Examples [ edit ] For example, if A = F 2 {\displaystyle A=\mathbb {F} _{2}} and n = 3 {\displaystyle n=3} , the set of codewords contained in cyclic code generated by ( 1 , 1 , 0 ...