In this article, for the finite field $\mathbb{F}_q$, we show that the $\mathbb{F}_q$-algebra $\mathbb{F}_q[x]/\langle f(x) \rangle$ is isomorphic to the product ring $\mathbb{F}_q^{\deg f(x)}$ if and only if $f(x)$ splits over $\mathbb{F}_q$ into distinct factors. We generalize this result to the quotient of the polynomial algebra $\mathbb{F}_q[x_1, x_2,\dots, x_k]$ by the ideal $\langle f_1(x_1), f_2(x_2),\dots, f_k(x_k)\rangle.$ On the other hand, every finite dimensional $\mathbb{F}_q$-algebra $\mathcal{A}$ has an orthogonal basis of idempotents with their sum equal to $1_{\mathcal{A}}$ if and only if $\mathcal{A}\cong\mathbb{F}_q^l$ as $\mathbb{F}_q$-algebras, where $l=\dim_{\mathbb{F}_q} \mathcal{A}$. We utilize this characterization to study polycyclic codes over $\mathcal{A}$ and get a unique decomposition of polycyclic codes over $\mathcal{A}$ into polycyclic codes over $\mathbb{F}_q$ for every such orthogonal basis of $\mathcal{A}$, which is referred to as an $\mathbb{F}_q$-decomposition. An $\mathbb{F}_q$-decomposition enables us to use results of polycyclic codes over $\mathbb{F}_q$ to study polycyclic codes over $\mathcal{A}$; for instance, we show that the annihilator dual of a polycyclic code over $\mathcal{A}$ is a polycyclic code over $\mathcal{A}$. Furthermore, we consider the obvious Gray map (which is obtained by restricting scalars from $\mathcal{A}$ to $\mathbb{F}_q$) to find and study codes over $\mathbb{F}_q$ from codes over $\mathcal{A}$. Finally, with the help of different Gray maps, we produce a good number of examples of MDS or almost-MDS or/and optimal codes; some of them are LCD over $\mathbb{F}_q$.

翻译：本文针对有限域 $\mathbb{F}_q$，证明了 $\mathbb{F}_q$-代数 $\mathbb{F}_q[x]/\langle f(x) \rangle$ 同构于积环 $\mathbb{F}_q^{\deg f(x)}$ 当且仅当 $f(x)$ 在 $\mathbb{F}_q$ 上分裂为互异因子。我们将此结果推广到多项式代数 $\mathbb{F}_q[x_1, x_2,\dots, x_k]$ 模理想 $\langle f_1(x_1), f_2(x_2),\dots, f_k(x_k)\rangle$ 的商代数。另一方面，每个有限维 $\mathbb{F}_q$-代数 $\mathcal{A}$ 存在一个幂等元正交基且其和等于 $1_{\mathcal{A}}$ 当且仅当 $\mathcal{A}\cong\mathbb{F}_q^l$ 作为 $\mathbb{F}_q$-代数，其中 $l=\dim_{\mathbb{F}_q} \mathcal{A}$。我们利用这一刻画研究 $\mathcal{A}$ 上的多循环码，并针对 $\mathcal{A}$ 的每个此类正交基（称为 $\mathbb{F}_q$-分解），得到 $\mathcal{A}$ 上多循环码到 $\mathbb{F}_q$ 上多循环码的唯一分解。$\mathbb{F}_q$-分解使我们能够利用 $\mathbb{F}_q$ 上多循环码的结果研究 $\mathcal{A}$ 上的多循环码；例如，我们证明 $\mathcal{A}$ 上多循环码的零化子对偶码仍是 $\mathcal{A}$ 上的多循环码。此外，我们通过显式 Gray 映射（通过将标量限制从 $\mathcal{A}$ 到 $\mathbb{F}_q$ 得到）来寻找和研究从 $\mathcal{A}$ 上码导出的 $\mathbb{F}_q$ 上码。最后，借助不同的 Gray 映射，我们构造了大量 MDS 码、几乎 MDS 码和/或最优码的实例；其中部分码在 $\mathbb{F}_q$ 上是 LCD 码。