Title: On cyclic 𝔽_q-linear 𝔽_q^t-codes

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 𝔽_q 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 𝔽ⁿ_q^t , 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 𝔽ⁿ_q^t for any prime power q and is alternating when q is even. We study dual codes of cyclic 𝔽_q-linear 𝔽_q^t-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 𝔽_q-linear 𝔽_q²-codes of length n, and enumerate all the self-orthogonal and self-dual cyclic 𝔽_q-linear 𝔽_q^t-codes of length n. Besides this, by placing ordinary and Hermitian trace inner products on 𝔽ⁿ_q², we determine bases of all the complementary-dual cyclic 𝔽_q-linear 𝔽_q²-codes of length n.

Keywords: sesquilinear forms; totally isotropic spaces; Witt index.

DOI: 10.1504/IJICOT.2017.081457

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: 09 Jan 2017 *

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