Fq-linear Cyclic Codes over $$F{_q^m}$$ : DFT Approach

Abstract

Codes over Fqm that are closed under addition, and multiplication with elements from Fq are called Fq-linear codes over Fqm. For m ≠ 1, this class of codes is a subclass of nonlinear codes. Among Fq-linear codes, we consider only cyclic codes and call them Fq-linear cyclic codes (FqLC codes) over Fqm. The class of FqLC codes includes as special cases (i) group cyclic codes over elementary abelian groups (q = p, a prime), (ii) subspace subcodes of Reed-Solomon codes (n = qm - 1) studied by Hattori, McEliece and Solomon, (iii) linear cyclic codes over Fq (m = 1) and (iv) twisted BCH codes. Moreover, with respect to any particular Fq-basis of Fqm, any FqLC code over Fqm can be viewed as an m-quasi-cyclic code of length mn over Fq. In this correspondence, we obtain transform domain characterization of FqLC codes, using Discrete Fourier Transform (DFT) over an extension field of Fqm. The characterization is in terms of any decomposition of the code into certain subcodes and linearized polynomials over Fqm. We show how one can use this transform domain characterization to obtain a minimum distance bound for the corresponding quasi-cyclic code. We also prove nonexistence of self dual FqLC codes and self dual quasi-cyclic codes of certain parameters using the transform domain characterization.