Structural properties and enumeration of quasi cyclic codes

Given any finite fieldF_q, an (N, K) quasi cyclic code is defined as aK dimensional linear subspace ofF_q^N which is invariant underTⁿfor some integern, 0 <n N, and whereT is the cyclic shift operator. Quasi cyclic codes are shown to be isomorphic to theF_q[]-submodules ofF_q^N where the product(gl)· is naturally defined as₀+₁Tⁿ+...+_mT^mnif()= ₀+₁+...+_mm.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 cyclicF_q[]-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.