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 $R_{u^2,v^2,p^m}[x;\mathrm{\Theta }],$ where $\mathrm{\Theta }$ 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 $x^{n{p}^s}-\lambda$, where $\gcd (n,p)=1$ and $\mathrm{\Theta }(\lambda )=\lambda$. This reduces the study of skew $(\lambda ,\mathrm{\Theta })$-constacyclic codes of length $n{p}^s$ to the study of left ideals of $\frac{R_{u^2,v^2,p^m}[x;\mathrm{\Theta }]}{⟨f(x{)}^j⟩},$ where $f(x)$ is a central irreducible divisor of degree $l$ of $x^{n{p}^s}-\lambda$, 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 $(\lambda ,\mathrm{\Theta })$-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 $\mathrm{\Theta }$ is the identity automorphism, we study constacyclic codes of length $n{p}^s$ 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 $n{p}^s$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$ to the finite non-chain ring $R_{u^2,v^2,p^m}$.