Structural properties and enumeration of quasi cyclic codes

Abstract

Abstract. Given any finite field Fq, an (N, K) quasi cyclic code is defined as a K dimensional linear subspace of Fq u which is invariant under T" for some integer n, 0 < n =< N, and where T is the cyclic shift operator. Quasi cyclic codes are shown to be isomorphic to the Fq[2]-submodules of Fq u where the product #(2).v is naturally defined as #o v + #1vT" + ... + #,,vT m" if #(2) = #o + #1 ' ;~ + "" + #,, 2m. 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 cyclic Fq[2]-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.

DFT Domain Characterization of Quasi-Cyclic Codes

“…Just taking a minimal family of such minimal subcodes such that their sum is the original code, we can express the code as the direct sum of some minimal m-quasi-cyclic codes. So we have Theorem 4.1 was first proved in [4]. Note that the decomposition of an mquasi-cyclic code in terms of some minimal m-quasi-cyclic codes may not be unique, though for m = 1, that is for cyclic codes the decomposition is always unique.…”

“…There are fifteen minimal α 0 = 1-invariant subspaces, each consisting of the zero element and any one nonzero element of F 2 4 . For any other value of s there is only one minimal s-invariant subspace which is F 2 4 .…”

“…In the case of cyclic codes the transform components from two different 4]. Notice that when m = 1, the q-cyclotomic cosets modulo n and the q-cyclotomic cosets modulo n m are identical and there is no room to relate transform components of different q-cyclotomic cosets.…”

“…The unique subcodes C i in (4), obtained by considering each q-cyclotomic coset modulo n m are actually the primary components [15] or irreducible components [4] of the code. In [15], the primary components of C were obtained as .…”

“…The authors then investigate the structural properties of m-quasi-cyclic codes with the help of Groebner bases of modules over F q [X]. Essentially the same module structure was imposed by Conan and Seguin in [4,25] in the unblocked forms of the codewords. They imposed an F q [X]-module structure on an m-quasi-cyclic code by defining f (X).a = f (T m )(a), where T is the cyclic shift operator.…”