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

Let Fq be a finite field of cardinality q, where q is a power of a prime number p, t ≥ 2 an even number satisfying t 6≡ 1 (mod p) and Fqt an extension field of Fq with degree t. First, a new trace bilinear form on F n qt which is called ∆-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 ∆-self-orthogonal and cyclic ∆-self-dual Fq-linear Fqt -codes are investigated when t = 2. Furthermore, some good Fq-linear Fq2 -codes are obtained.