Classification of a specific class 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 an explicit classification, up to isomorphism, of the commutative, $3$-nilpotent $\mathbb{F}_{p^k}$-algebras $(V,+,\cdot)$ with $\dim(V\cdot V)=1$, starting from the connection with their bi-brace structure. Such classification is the generalization in odd characteristic of the results obtained by Civino at al.\ in characteristic $2$. The study of this particular case arises from the recent interest raised by cryptography research for its application in the cryptanalysis of block ciphers.