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

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 ?_i=1^lm_i{\sum_{i=1}^lm_i} is an Rx]-submodule of Rx]/(x^m₁-1)×?×Rx]/(x^m_l-1){Rx]/(x^{m_1}-1)\times\cdots \times Rx]/(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{x^{m_i}-1} (1 ≤ i ≤ l). Then the GQC codes over R of block lengths (m ₁, m ₂, . . . , m _l ) and length ?_i=1^lm_i{\sum_{i=1}^lm_i} are identified with G₁×?×G_t{{\mathcal G}_1\times\cdots\times {\mathcal G}_t}, where G_j{{\mathcal G}_j} is an Rx]/(g _j )-submodule of (Rx]/(g_j))^n_j{(Rx]/(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{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.