伽罗华环上λ-循环码的结构

一些重要的二元非线性码是Z4上线性码在Glay映射下的像集,因而需要对有限环上的线性码特别是循环码的研究给予特别关注.设p是素数,R=GR（ps,pms）是特征为ps并且元素个数为psm的Galois环,选定λ∈R并且λ是非零因子.设C是R上的长为n的线性码,如果c=（c0,c1,…,cn-1）∈C都有（λcn-1,c0,c1,…,cn-2）∈C,则称是R上长为n的λ-循环码.R上的λ-循环码可以等同于商环Rλn=Rx]/〈xn-λ〉中的理想.设xn-λ=f1…fk,fi=（xn-λ）/fi,其中f1,…,fk是R上两两互素,首项系数为1的基本不可约多项式,证明了Rλn中的任何理想都是形如〈pj fi＋〈xn-λ〉〉的一些理想的内直和,其中0≤j≤s,1≤i≤k;Rλn共有（s＋1）k个理想;Rx]/〈xn-λ〉是主理想环.

The Structure of λ-Cyclic Codes over Galois Rings

Important families of binary non-linear codes are image under a Glay map of linear codes of over Z 4.Therefore it is of interest to study linear codes,in particular cyclic codes,over finite rings.Let pbe a prime,R=GR(p s,p ms) a Galois ring of characteristic p s and cardinality p sm,λ∈Rnon-zero divisor,Let Cbe linear code with length nover R.Cis calledλ-cyclic with length nover Rif whenever c=(c 0,c 1,…,c n-1) ∈C,(λc n-1,c 0,c 1,…,c n-2) ∈C.λ-cyclic codes over Rcan be regarded as ideals of the quotient ring R λ n =R x]/〈 x n-λ 〉.Let x n-λ=f 1 … fk,fi =(x n-λ)/ fi,where f 1,… fk are pairwise coprime monic basic irreducible polynomials over R.It is proven that any ideal of R λ n is a direct sum of ideals of the type 〈 pjfi + 〈 x n-λ 〉〉,where 0≤j≤s,1≤i≤k;the number of ideals of R λ n is(s+1) k;R x /〈xn-λ 〉 is a principal ideal ring.