Generalized Cyclic Codes over Product Ring PQ
- 1Department of Mathematics, College of Science, Taibah University, Madinah 42353, Saudi Arabia
- 2Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah 42351, Saudi Arabia
Abstract
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.
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BibTeX
@article{almuhaimeed2026generalized,
title = {{Generalized Cyclic Codes over Product Ring PQ}},
author = {Areej Almuhaimeed and Abu Zaid Ansari and Faiza Shujat},
journal = {Axioms},
year = {2026},
month = oct,
doi = {10.3390/axioms15100772},
url = {https://www.mdpi.com/2075-1680/15/10/772},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Generalized Cyclic Codes over Product Ring PQ
AU - Areej Almuhaimeed
AU - Abu Zaid Ansari
AU - Faiza Shujat
PY - 2026
DA - 2026/10/15
JO - Axioms
DO - 10.3390/axioms15100772
UR - https://www.mdpi.com/2075-1680/15/10/772
AB - 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.
SN - 2075-1680
ER -