Generalized Cyclic Codes over Product Ring PQ

This paper investigates cyclic codes over Fqn×Pm×Qk, where Fq is a finite field of odd characteristic, P=Fq[v]/⟨v2−τ2v⟩, and Q=Fq[u,v]/⟨u3−τ12u,v2−τ2v⟩, with τ1,τ2∈Fq*. Using complete systems of orthogonal idempotents, we decompose (P) and (Q) into two and six constituent codes over Fq, respectively, and define matrix-based Gray maps. The induced Gray map from Fqn×Pm×Qk to Fqn+2m+6k is shown to be Fq-linear and Lee–Hamming distance preserving. We prove that a mixed-alphabet code is cyclic if and only if its deinterleaved Gray image is a generalized quasi-cyclic code with block-length profile (n, m, m, k, k, k, k, k, k). A polynomial-module characterization, constituent generator polynomials, dimension bounds, shift-generated spanning sets, and structural results for the Euclidean dual and hull are also established. Furthermore, the generator structure and cardinality of cyclic codes over P, Q, and the mixed alphabet are determined through their field constituents. As an application, scalar-orthogonal Gray matrices and dual-containing constituent cyclic codes are combined with the Calderbank–Shor–Steane construction. An explicit construction over F5 yields a dual-containing generalized quasi-cyclic code with parameters [90, 75, 3]5, and a quantum error-correcting code with parameters [[90, 60, 3]]5.