Quantum Codes from Group Ring Codes

TL;DR

This paper explores the construction and analysis of quantum codes using group ring codes over various finite groups, providing a unified algebraic framework and conditions for self-orthogonality essential for quantum error correction.

Contribution

It introduces a comprehensive group ring approach for quantum code construction, extending to complex group structures and establishing criteria for self-orthogonality.

Findings

Unified algebraic framework for quantum codes

Conditions for self-orthogonality under various inner products

Extension to complex group structures like dihedral and semidirect products

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.