On cyclic 𝔽<SUB align="right">q-linear 𝔽<SUB align="right">q<SUP align="right">t</SUP>-codes

Abstract

Let 𝔽q denote the finite field of order q and characteristic p, n be a positive integer coprime to q and t ≥ 2 be an integer satisfying t ≢ 1(mod p). In this paper, we place a new trace bilinear form on 𝔽nqt , which is called the * trace bilinear form and is a generalisation of the trace inner product when q = t = 2 and Hermitian trace inner product when q is even and t = 2. We observe that it is a non-degenerate, symmetric bilinear form on 𝔽nqt for any prime power q and is alternating when q is even. We study dual codes of cyclic 𝔽q-linear 𝔽qt-codes of length n with respect to this bilinear form. We also explicitly determine bases of all the complementary-dual, self-orthogonal and self-dual cyclic 𝔽q-linear 𝔽q2-codes of length n, and enumerate all the self-orthogonal and self-dual cyclic 𝔽q-linear 𝔽qt-codes of length n. Besides this, by placing ordinary and Hermitian trace inner products on 𝔽nq2, we determine bases of all the complementary-dual cyclic 𝔽q-linear 𝔽q2-codes of length n.