A new trace bilinear form on cyclic $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes

TL;DR

This paper introduces a novel trace bilinear form called the elta-bilinear form on cyclic elta-self-orthogonal and self-dual codes over finite field extensions, providing new classifications and constructions for these codes.

Contribution

It proposes a new elta-bilinear form on _{q^t}^n and explores bases, enumeration, and construction of cyclic elta-self-orthogonal and self-dual codes for t=2.

Findings

Classification of cyclic elta-self-orthogonal codes

Enumeration of elta-self-dual codes

Construction of good _q-linear _{q^2}-codes

Abstract

Let $\mathbb{F}_q$ be a finite field of cardinality $q$, where $q$ is a power of a prime number $p$, $t\geq 2$ an even number satisfying $t \not\equiv 1 \;(\bmod \;p)$ and $\mathbb{F}_{q^t}$ an extension field of $\mathbb{F}_q$ with degree $t$. First, a new trace bilinear form on $\mathbb{F}_{{q^t}}^n$ which is called $\Delta$-bilinear form is given, where $n$ is a positive integer coprime to $q$. Then according to this new trace bilinear form, bases and enumeration of cyclic $\Delta$-self-orthogonal and cyclic $\Delta$-self-dual $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes are investigated when $t=2$. Furthermore, some good $\mathbb{F}_q$-linear $\mathbb{F}_{q^2}$-codes are obtained.