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

abstract

Codes over $F{_q{^m}}$ that are closed under addition, and multiplication with elements from F _q are called F _q -linear codes over $F_{q{^m}}$. For m≠ 1, this class of codes is a subclass of nonlinear codes. Among F _q -linear codes, we consider only cyclic codes and call them F _q -linear cyclic codes (F _q LC codes) over $F_{q{^m}}$ The class of F _q LC 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=q ^m−1) studied by Hattori, McEliece and Solomon, (iii) linear cyclic codes over F _q (m=1) and (iv) twisted BCH codes. Moreover, with respect to any particular F _q -basis of $F_{q{^m}}$, any F _q LC code over $F_{q{^m}}$ can be viewed as an m-quasi-cyclic code of length mn over F _q . In this correspondence, we obtain transform domain characterization of F _q LC codes, using Discrete Fourier Transform (DFT) over an extension field of $F_{q{^m}}$ The characterization is in terms of any decomposition of the code into certain subcodes and linearized polynomials over $F_{q{^m}}$. 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 F _q LC codes and self dual quasi-cyclic codes of certain parameters using the transform domain characterization.