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

Abstract

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 $R_{u^2, v^2, p^m}[x; Θ],$ where $Θ$ 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 \(x^{np^s}-λ\), where \(\gcd(n,p)=1\) and \(Θ(λ)=λ\). This reduces the study of skew \((λ,Θ)\)-constacyclic codes of length \(np^s\) to the study of left ideals of $\frac{R_{u^2,v^2,p^m}[x;Θ]}{\langle f(x)^j\rangle}, $ where \(f(x)\) is a central irreducible divisor of degree \(l\) of \(x^{np^s}-λ\), for an invertible element \(λ\in R_{u^2,v^2,p^m}\) and \(j\in\mathbb{N}\). We then apply these results to skew \((λ,Θ)\)-constacyclic codes of length \(p^s\) for different classes of units \(λ\). Several examples are presented to illustrate the theory and to obtain optimal codes. Finally, when \(Θ\) is the identity automorphism, we study constacyclic codes of length \(np^s\) over \(R_{u^2,v^2,p^m}\), according as \(x^n-α_0\) 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 \(R_{u^2,v^2,p^m}\).