  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 $A_{H_8}$ is an axial algebra and embeds in the code VOA $V_{H_8}$.
