A classification of ${\mathbb{F}}_{p^k}$-braces using bilinear forms

Let ${\mathbb{F}}_{p^k}$ be a finite field of odd characteristic $p$. In this paper we give a classification, up to isomorphism, of the associative commutative ${\mathbb{F}}_{p^k}$-algebras, starting from the connection with their bi-brace structure. Such classification is the generalization in odd characteristic of the result proved by Civino at al. in characteristic $2$.