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

A code is s-quasi-cyclic (s-QC) if there is an integer s such that cyclic shift of a codeword by s-positions is also a codeword. For s = 1, cyclic codes are obtained. A dyadic code is a code which is closed under all dyadic shifts. An s-QC dyadic (s-QCD) code is one which is both s-QC and dyadic. QCD codes with s = 1 give codes that are cyclic and dyadic (CD). 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 s-QCD code is also an s-QCD code and s-QCD codes for a given dimension are enumerated for all possible values of s.