On the Algebraic Structure of Quasi-cyclic Codes IV: Repeated Roots

A trace formula for quasi-cyclic codes over rings of characteristic not coprime with the co-index is derived. The main working tool is the Generalized Discrete Fourier Transform (GDFT), which in turn relies on the Hasse derivative of polynomials. A characterization of Type II self-dual quasi-cyclic codes of singly even co-index over finite fields of even characteristic follows. Implications for generator theory are shown. Explicit expressions for the combinatorial duocubic, duoquintic and duoseptic constructions in characteristic two over finite fields are given. AMS Classification: 94B05, 94B15, 11T71 Part of this work was done while the first named author was visiting CNRS-I3S, ESSI, Sophia Antipolis, France. The author would like to thank the institution for the kind hospitality. The research of the first two authors is partially supported by MOE-ARF research Grant R-146-000-029-112 and DSTA research Grant R-394-000-011-422.