Axial algebras are a recently introduced class of non-associative algebra motivated by applications to groups and vertex-operator algebras. We develop the structure theory of axial algebras focussing on two major topics: (1) radical and simplicity; and (2) sum decompositions.

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 .

An axial algebra A is a commutative non-associative algebra generated by primitive idempotents, called axes, whose adjoint action on A is semisimple and multiplication of eigenvectors is controlled by a certain fusion law. Different fusion laws define different classes of axial algebras.Axial algebras are inherently related to groups. Namely, when the fusion law is graded by an abelian group T , every axis a leads to a subgroup of automorphisms T a of A. The group generated by all T a is called the Miyamoto group of the algebra. We describe a new algorithm for constructing axial algebras with a given Miyamoto group.A key feature of the algorithm is the expansion step, which allows us to overcome the 2-closeness restriction of Seress's algorithm computing Majorana algebras.At the end we provide a list of examples for the Monster fusion law, computed using a magma implementation of our algorithm.