Structural properties and enumeration of quasi cyclic codes
Structural properties and enumeration of quasi cyclic codes
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DOI：
10.1007/bf01270398
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发表时间：
1993-03
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通讯作者：
J. Conan;G. Séguin
中科院分区：
文献类型：
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作者：
J. Conan;G. Séguin
给定任意有限域Fq，(N，K)拟循环码被定义为FqN的K维线性子空间，它对于某个整数n，0<；n≦N在Tn下是不变的，其中T是循环移位算子。证明了拟循环码与FqN的Fq[λ]-子模同构，其中乘积μ(Gl)·ν自然定义为μ0ν+μ1νTn+...+μmνTmnifμ(λ)=μ0+μ1+...+μmλm.在(N/n，q)=1的情况下，证明了所有拟循环码都可分解成固定数目的不可分解分量的直和，称为不可约循环Fq[λ]-子模，给出了包括循环码、具有循环基的拟循环码、极大和不可约循环码在内的某些子类的完全特征和计数.最后给出了一个确定和刻画任意拟循环码的对偶的一般方法。
Given any finite fieldFq, an (N, K) quasi cyclic code is defined as aKdimensional linear subspace ofFqNwhich is invariant underTnfor some integern, 0 <n≦N, and whereTis the cyclic shift operator. Quasi cyclic codes are shown to be isomorphic to theFq[λ]-submodules ofFqNwhere the productμ(gl)·νis naturally defined asμ0ν+μ1νTn+...+μmνTmnifμ(λ)= μ0+μ1+...+μmλm.In the case where (N/n, q)=1, all quasi cyclic codes are shown to be decomposable into the direct sum of a fixed number of indecomposable components called irreducible cyclicFq[λ]-submodules providing for the complete characterisation and enumeration of some subclasses of quasi cyclic codes including the cyclic codes, the quasi cyclic codes with a cyclic basis, the maximal and the irreducible ones. Finally a general procedure is presented which allows for the determination and characterisation of the dual of any quasi cyclic code.