Authors: Anuradha Sharma; Taranjot Kaur
Addresses: Center for Applied Mathematics, IIIT Delhi, New Delhi, India ' Department of Mathematics, IIT Delhi, Hauz Khas, New Delhi, India
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.
Keywords: sesquilinear forms; totally isotropic spaces; Witt index.
International Journal of Information and Coding Theory, 2017 Vol.4 No.1, pp.19 - 46
Received: 12 Apr 2016
Accepted: 29 Jul 2016
Published online: 02 Jan 2017 *