The binary Reed-Muller code evaluates Boolean polynomials of degree at most on . It has parametersThe dimension counts square-free monomials, and the minimum weight follows inductively from the decomposition .
If have minimum distances , respectively, thenIndeed, , while the words and attain the two candidate bounds.
If is a parity-check matrix of , then a parity-check matrix of isIts equations say that the first half lies in and that the sum of the two halves lies in .
Under the standard binary inner product,The product of representing polynomials has degree at most , so its evaluation word has even weight; equality follows by comparing dimensions.
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Reed–Muller codes are a family of error-correcting codes that are used in digital communication and data storage to detect and correct errors in transmitted or stored data. They are particularly known for their simple decoding algorithms and their good performance in terms of error correction capabilities.