## Abstract

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 C be a binary linear code of length n. A basis for the code algebra A_{C} consists of n idempotents 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 H_{8} where the code algebra A_{H8
} is an axial algebra and embeds in the code VOA V_{H8
}.

Original language | English |
---|---|

Pages (from-to) | 146-176 |

Number of pages | 31 |

Journal | Journal of Algebra |

Volume | 518 |

Early online date | 22 Oct 2018 |

DOIs | |

Publication status | Published - 15 Jan 2019 |

## Keywords

- Axial algebras
- Binary linear codes
- Nonassociative algebras
- Vertex operator algebras