Polyadic cyclic codes over a non-chain ring $$\mathbb{F}_{q}[u,v]/\langle f(u),g(v), uv-vu\rangle$$

Abstract: Let $f(u)$ and $g(v)$ be any two polynomials of degree $k$ and $\ell$ respectively ($k$ and $\ell$ are not both $1$), which split into distinct linear factors over $\mathbb{F}<em>{q}$. Let $$\mathcal{R}=\mathbb{F}</em>{q}[u,v]/\langle f(u),g(v),uv-vu\rangle$$ be a finite commutative non-chain ring. In this paper, we study polyadic codes and their extensions over the ring $\mathcal{R}$. We give examples of some polyadic codes which are optimal with respect to Griesmer type bound for rings. A Gray map is defined from $$\mathcal{R}ⁿ \rightarrow \mathbb{F}^{k\ell n}_q$$ which preserves duality. The Gray images of polyadic codes and their extensions over the ring $\mathcal{R}$ lead to construction of self-dual, isodual, self-orthogonal and complementary dual (LCD) codes over $\mathbb{F}_q$. Some examples are also given to illustrate this.