Hermitian Self-Orthogonal Constacyclic Codes over Finite Fields

Necessary and sufficient conditions for the existence of Hermitian self-orthogonal constacyclic codes of length over a finite field , coprime to , are found. The defining sets and corresponding generator polynomials of these codes are also characterised. A formula for the number of Hermitian self-orthogonal constacyclic codes of length over a finite field is obtained. Conditions for the existence of numerous MDS Hermitian self-orthogonal constacyclic codes are obtained. The defining set and the number of such MDS codes are also found. 1. Introduction Let denote a finite field with elements. An linear code of length and dimension over is a -dimensional subspace of the vector space . Elements of the subspace are called codewords and are written as row vectors . A linear code over is called -constacyclic if is in for every in . Let be the map given by . One can easily check that is an -module isomorphism. We can therefore identify -constacyclic codes of length over with ideals in . The Hamming weight of is the number of nonzero coordinates of . The minimum distance of is defined to be . An code, that is, a linear code with minimum distance , is said to be maximum distance separable (MDS) if . The Hermitian inner product of elements is defined as , for and . For a linear code of length over , the Hermitian dual code of is defined by . If , then is known as Hermitian self-dual and is Hermitian self-orthogonal if . Aydin et al. [1] dealt with constacyclic codes and a constacyclic BCH bound was given. Gulliver et al. [2] showed that there exists Euclidean self-dual MDS code of length over when by using a Reed-Solomon (RS) code and its extension. They also constructed many new Euclidean and Hermitian self-dual MDS codes over finite fields. Blackford [3] studied negacyclic codes over finite fields by using multipliers. He gave conditions on the existence of Euclidean self-dual codes. Recently, Guenda [4] constructed MDS Euclidean and Hermitian self-dual codes from extended cyclic duadic or negacyclic codes and gave necessary and sufficient conditions on the existence of Hermitian self-dual negacyclic codes arising from negacyclic codes. In [5] the authors gave formulae to enumerate the number of Euclidean self-dual and self-orthogonal negacyclic codes of length over a finite field , where is coprime to . In [6] Yang and Cai gave the necessary and sufficient conditions for the existence of Hermitian self-dual constacyclic codes. They also gave some conditions under which Hermitian self-dual and self-orthogonal MDS codes exist. In this paper, we find necessary and