A Study Of Skew-Polycyclic Codes Over A Non-Chain Ring

For a prime $p$ and a positive integer $m$, let $\mathbb{F}_{p^m}$ be the finite field of cardinality $p^m$, and let $R_{u^2,v^2,p^m} =\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+v\mathbb{F}_{p^m} +uv\mathbb{F}_{p^m}, ~ u^2=v^2=0,\ uv=vu,$ be a finite non-chain ring. In this paper, we study skew polycyclic codes of length $lj$ associated with $f(x)^j$, where $f(x)$ is a central polynomial of degree $l$ in $R_{u^2, v^2, p^m}[x; \Theta],$ where $\Theta$ being an automorphism of $R_{u^2,v^2,p^m}$. We describe these codes, characterize free skew polycyclic codes, and determine their ranks. Under suitable centrality assumptions, we decompose the quotient ring associated with $x^{np^s}-\lambda$, where $\gcd(n,p)=1$ and $\Theta(\lambda)=\lambda$. This reduces the study of skew $(\lambda,\Theta)$-constacyclic codes of length $np^s$ to the study of left ideals of $\frac{R_{u^2,v^2,p^m}[x;\Theta]}{\langle f(x)^j\rangle},$ where $f(x)$ is a central irreducible divisor of degree $l$ of $x^{np^s}-\lambda$, for an invertible element $\lambda\in R_{u^2,v^2,p^m}$ and $j\in\mathbb{N}$. We then apply these results to skew $(\lambda,\Theta)$-constacyclic codes of length $p^s$ for different classes of units $\lambda$. Several examples are presented to illustrate the theory and to obtain optimal codes. Finally, when $\Theta$ is the identity automorphism, we study constacyclic codes of length $np^s$ over $R_{u^2,v^2,p^m}$, according as $x^n-\alpha_0$ is irreducible or reducible over $\mathbb{F}_{p^m}$. These results extend the work of \cite{CCDF18} and \cite{ZTG18} on constacyclic codes of length $np^s$ over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$ to the finite non-chain ring $R_{u^2,v^2,p^m}$.