Generalizations of Cyclic Codes over Product Rings

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 $\mathbb{F}_q$.