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 $p$, $t\geq 2$ 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.