Code algebras which are axial algebras and their $\mathbb{Z}_2$-gradings

TL;DR

This paper explores the structure of code algebras derived from binary codes, focusing on small idempotents, their properties, and conditions under which these algebras are axial and possess non-trivial $bZ_2$-gradings, including new examples.

Contribution

It introduces the concept of small idempotents in code algebras, proves their properties, and classifies when the algebra admits a $bZ_2$-grading, including the first examples of non-trivial gradings.

Findings

Small idempotents are primitive and semisimple.

Code algebras generated by small idempotents are axial.

Infinite family of $bZ_2 imes bZ_2$-graded axial algebras discovered.

Abstract

A code algebra $A_C$ is a non-associative commutative algebra defined via a binary linear code $C$. We study certain idempotents in code algebras, which we call small idempotents, that are determined by a single non-zero codeword. For a general code $C$, we show that small idempotents are primitive and semisimple and we calculate their fusion law. If $C$ is a projective code generated by a conjugacy class of codewords, we show that $A_C$ is generated by small idempotents and so is, in fact, an axial algebra. Furthermore, we classify when the fusion law is $\mathbb{Z}_2$-graded. In doing so, we exhibit an infinite family of $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded axial algebras - these are the first known examples of axial algebras with a non-trivial grading other than a $\mathbb{Z}_2$-grading.