| There are researches on QC codes in 1960s.Recently,Kasami showed that QC codes meet a modified Gilbert-Varshamov bound. As a result ,the research on QC codes attracts many scholars' attention again.In the second chapter of this paper,we discuss the 1-generator quasi-cyclic over finite filed IFq. The 1-generator quasi-cyclic codes and their duals are the most frequently encountered QC codes in the literature. Séguin discusses the construction and the enumeration of 1-generator QC codes in the special case where the prime factorization of xm — 1 is the same in IFqn[x] as in IFq[x]. In this paper,we will discuss the construction and the enumeration of 1-generator QC codes in the general case, and describe an algorithm which will obtain one,and only one,generator for each 1-generator QC codes.In the third chapter of this paper,we discuss that the quasi-cyclic codes over (?)4, Quasi-cyclic codes of length mm over Z4 are equivalent to A-submodules of An,where A = (?)4[x]/(xm — 1). In the case of m being odd ,all quasi-cyclic codes are shown to be decomposable into the direct sum of a fixed number of cyclic irreducible A-submodules. Furthermore,the decompositions are used to emimerate the distinct quasi-cyclic as well as some specific subclasses.Finally a general procedure which allows for determining the dual of any quasi-cyclic code is presented. |
