指标为3的自对偶拟循环码

Self-dual Quasi-cyclic Codes of Index 3

设l为非零自然数,R=Fq[x]/〈xm-1〉,这里Fq为有限域.视拟循环码为代数Rl上的一个子模,利用模上的Grbner基理论及拟循环码的代数结构作为工具,得到了两个主要定理:在l=3的情况下,把一个关于rPOT项序的Grbner基生成集转化为一个关于POT项序的既约Grbner基生成集;指标为3的拟循环码是自对偶码的充要条件.

Let g be a nonzero natural number, and let R=Fq[x]/〈x^m-1〉, where Fq is a finite field. Regard a quasi-cyclic code as a sub-module of the algebra Re, using Grobner bases of modules and algebraic structure of quasi-cyclic codes as tools, and then we get two main theorems as follows, the way to have a formula, in the index 3 case, for the POT Grobner bases generating set in terms of a Grobner bases（rPOT） generating set; the necessary and sufficient condition on which quasi-cyclic codes of index 3 are self-dual, which provides a complete characterization of self-dual quasi-cyclic codes of index 3.