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

Codes over that are closed under addition, and multiplication with elements from F_q are called F_q-linear codes over . 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 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 , any F_qLC code over 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 The characterization is in terms of any decomposition of the code into certain subcodes and linearized polynomials over . 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.AMS classification 94B05