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Let $\mathbb{F}_q$ be a finite field of cardinality $q$, where $q$ is a powerof 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 $\mathbb{F}_q$ withdegree $t$. First, a new trace bilinear form on $\mathbb{F}_{{q^t}}^n$ which iscalled $\Delta$-bilinear form is given, where $n$ is a positive integer coprimeto $q$. Then according to this new trace bilinear form, bases and enumerationof 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 areobtained.

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