$\mathbb{F}$-valued trace of a finite-dimensional commutative $\mathbb{F}$-algebra

TL;DR

This paper identifies classes of finite-dimensional commutative algebras over a field that admit an $F$-valued trace, constructs explicit trace maps, and explores their applications in algebraic coding theory.

Contribution

It explicitly constructs $F$-valued trace maps for certain finite-dimensional commutative algebras and discusses their theoretical and computational significance.

Findings

Existence of $F$-valued trace maps in an infinite class of algebras.

Construction of explicit trace maps for these algebras.

Potential applications in algebraic coding theory.

Abstract

A non-zero $\mathbb{F}$-valued $\mathbb{F}$-linear map on a finite dimensional $\mathbb{F}$-algebra is called an $\mathbb{F}$-valued trace if its kernel does not contain any non-zero ideals. However, given an $\mathbb{F}$-algebra such a map may not always exist. We find an infinite class of finite-dimensional commutative $\mathbb{F}$-algebras which admit an $\mathbb{F}$-valued trace. In fact, in these cases, we explicitly construct a trace map. The existence of an $\mathbb{F}$-valued trace on a finite dimensional commutative $\mathbb{F}$-algebra induces a non-degenerate bilinear form on the $\mathbb{F}$-algebra which may be helpful both theoretically and computationally. In this article, we suggest a couple of applications of an $\mathbb{F}$-valued trace map of an $\mathbb{F}$-algebra to algebraic coding theory.