Generators of negacyclic codes over \({\mathbb {F}}_{p}[u,v]/\langle u^2,v^2,uv,vu\rangle \) of length \(p^s\)

We consider a local non-Frobenius ring \(R_p={\mathbb {F}}_p[u,v]/\langle u^2, v^2, uv,vu \rangle \) $R_p={\mathbb{F}}_p[u,v]/⟨u^2,v^2,uv,vu⟩$ defined for each prime number p and study the ideal representation of the ring \(R_p[x]/\langle x^n+1\rangle \) $R_p[x]/⟨x^n+1⟩$ , which is well-known to be related to negacyclic codes over \(R_p\) $R_p$ of length \(n=p^s\) $n=p^s$ with \(s \ge 1\) $s\ge 1$ . We assume that n is a power of p, and under this setting, we show that a specific class of ideals can be represented uniquely in terms of degrees related to its generators. We also give a lower bound of the minimum Hamming distances of negacyclic codes over \(R_p\) $R_p$ , and show that the lower bound is sharp for several codes whose corresponding ideals of \(R_p[x]/\langle x^n+1\rangle \) $R_p[x]/⟨x^n+1⟩$ belong to the aforementioned class.