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

Abstract

Let ${\mathbb{F}}_q$ be a finite field of cardinality $q$, where $q$ is a power of a prime number $p$, $t\ge 2$ an even number satisfying $t\not\equiv 1(\mathrm{mod}p)$ 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.