Generalized quasi-cyclic codes over Galois rings: structural properties and enumeration

Abstract

For R a Galois ring and m ₁, . . . , m _l positive integers, a generalized quasi-cyclic (GQC) code over R of block lengths (m ₁, m ₂, . . . , m _l ) and length ${\sum_{i=1}^lm_i}$ is an R[x]-submodule of ${R[x]/(x^{m_1}-1)\times\cdots \times R[x]/(x^{m_l}-1)}$. Suppose m ₁, . . . , m _l are all coprime to the characteristic of R and let {g ₁, . . . , g _t } be the set of all monic basic irreducible polynomials in the factorizations of ${x^{m_i}-1}$ (1 ≤ i ≤ l). Then the GQC codes over R of block lengths (m ₁, m ₂, . . . , m _l ) and length ${\sum_{i=1}^lm_i}$ are identified with ${{\mathcal G}_1\times\cdots\times {\mathcal G}_t}$, where ${{\mathcal G}_j}$ is an R[x]/(g _j )-submodule of ${(R[x]/(g_j))^{n_j}}$, where n _j is the number of i for which g _j appears in the factorization of ${x^{m_i}-1}$ into monic basic irreducible polynomials. This identification then leads to an enumeration of such GQC codes. An analogous result is also obtained for the 1-generator GQC codes. A notion of a parity-check polynomial is given when R is a finite field, and the number of GQC codes with a given parity-check polynomial is determined. Finally, an algorithm is given to compute the number of GQC codes of given block lengths.