Quantum Codes from Group Ring Codes
Kanat Abdukhalikov, Tushar Bag, Daniel Panario
TL;DR
This work develops a group-ring approach to coding, unifying one- and two-dimensional cyclic codes through the algebra \\mathbb{F}_qG for various finite groups G (including \\mathbb{F}_qD_n, \\mathbb{F}_q(C_l \times C_m), and \\mathbb{F}_q(C_l \times D_m)). It provides explicit generator-matrix constructions using circulant and reverse-circulant blocks, derives self-orthogonality conditions under Euclidean, Hermitian, and symplectic inner products, and uses these to construct quantum error-correcting codes. The paper demonstrates both general theory (duality, dimensions, reciprocal polynomials) and concrete examples that yield competitive QECC parameters (often aligning with or matching best-known results). The framework offers flexibility to explore a broad class of group structures, potentially improving code parameters by selecting among multiple generator matrices for a fixed length. Overall, the results advance quantum code design by leveraging group-algebra structure to systematically obtain self-orthogonal codes and associated QECCs.
Abstract
This article examines group ring codes over finite fields and finite groups. We also present a section on two-dimensional cyclic codes in the quotient ring $\mathbb{F}_q[x, y] / \langle x^{l} - 1, y^{m} - 1 \rangle$. These two-dimensional cyclic codes can be analyzed using the group ring $\mathbb{F}_q(C_{l} \times C_{m})$, where $C_{l}$ and $C_{m}$ represent cyclic groups of orders $l$ and $m$, respectively. The aim is to show that studying group ring codes provides a more compact approach compared to the quotient ring method. We further extend this group ring framework to study codes over other group structures, such as the dihedral group, direct products of cyclic and dihedral groups, direct products of two cyclic groups, and semidirect products of two groups. Additionally, we explore necessary and sufficient conditions for such group ring codes to be self-orthogonal under Euclidean, Hermitian, and symplectic inner products and propose a construction for quantum codes.