Quasi-cyclic dyadic codes in the Walsh--Hadamard transform domain

A code is -quasi-cyclic ( -QC) if there is an integer such that cyclic shift of a codeword by -positions is also a codeword. For = 1, cyclic codes are obtained. A dyadic code is a code which is closed under all dyadic shifts. An -QC dyadic ( -QCD) code is one which is both -QC and dyadic. QCD codes with = 1 give codes that are cyclic and dyadic (CD). In this correspondence, we obtain a simple characterization of all QCD codes (hence of CD codes) over any field of odd characteristic using Walsh–Hadamard transform defined over that finite field. Also, it is shown that dual a code of an -QCD code is also an -QCD code and -QCD codes for a given dimension are enumerated for all possible values of .