Research On Quasi-cyclic Codes And Quasi-twisted Codes Over Some Classes Of Finite Chain Rings

The classic error-correcting codes, that is the error-correcting codes over thefinite field, had already been studied deeply. Many scholars and math lovers areputting their eyes on the error-correcting codes over the finite rings, especially thefinite chain rings. This article discusses the algebraic structure of the quasi-cycliccodes overZ_p^sand a class of quasi-twisted codes overR=F_p+uF_p （u²=0）1. Quasi-cyclic codes of length mn overZ_p^sare discussed and shown to beequivalent to A-submodules ofA nand study the type of distinct irreducibleA-submodules ofAⁿ, where A=Z_p^s[x]/（x^m-1）.When gcd（m, p）=1, all quasi-cyclic codes are shown to be decomposable into the direct sum of a fixed number ofcyclic irreducible A-submodules, and the necessary and sufficient condition thatquasi-cyclic code is irreducible submodule and cyclic module has been shown,respectively. Last, we enumerate the distinct quasi-cyclic codes.2. One class of quasi-twisted codes over the ringR=F_p+uF_p （u²=0）isdiscussed. First, the quasi-twisted codes over the ring R could be decomposed intodirect sum of irreducible module. Then, for the given decomposition we can get thedecomposition of its dual code. We also use the discrete Fourier transform to givethe inverse formula of the simple-root case. Then we produce a formula which canbe used to construct a QT code for the given component codes. Lastly, we study andconfirm the necessary and sufficient conditions that one class of self-dualquasi-twisted code overF₂+uF₂（u²=0）is TypeII code.