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.