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2018_EJRNL_PP_ALONSO_CASTILLO_1.pdf
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Inspired by code vertex operator algebras (VOAs) and their representation theory, we define code algebras, a new class of commutative non-associative algebras constructed from binary linear codes. Let Cbe a binary linear code of lengthn. A basis for the code algebra ACconsists of nidempotents and a vector for each non-constant codeword of C. We show that code algebras are almost always simple and, under mild conditions on their structure constants, admit an associating bilinear form. We determine the Peirce decomposition and the fusion law for the idempotents in the basis, and we give a construction to find additional idempotents, called the s-map, which comes from the code structure. For a general code algebra, we classify the eigenvalues and eigenvectors of the smallest examples of the s-map construction, and hence show that certain code algebras are axial algebras. We give some examples, including that for a Hamming code H8where the code algebra AH8is an axial algebra and embeds in the code VOA VH8.