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.

翻译：本文研究了有限域和有限群上的群环码。我们还讨论了商环 $\mathbb{F}_q[x, y] / \langle x^{l} - 1, y^{m} - 1 \rangle$ 中的二维循环码。这些二维循环码可以利用群环 $\mathbb{F}_q(C_{l} \times C_{m})$ 进行分析，其中 $C_{l}$ 和 $C_{m}$ 分别表示阶为 $l$ 和 $m$ 的循环群。本文旨在证明，与商环方法相比，研究群环码提供了一种更为简洁的途径。我们进一步扩展了这一群环框架，以研究其他群结构上的码，例如二面体群、循环群与二面体群的直积、两个循环群的直积以及两个群的半直积。此外，我们探讨了此类群环码在欧几里得内积、厄米内积和辛内积下自正交的充分必要条件，并提出了一种量子码的构造方法。