On $\mathbb{F}_{p^m}$ $ \mathfrak{R}$-additive skew negacyclic codes

Let \mathfrak{R} = \mathbb{F}_{p^m} + u\mathbb{F}_{p^m} with u^2 = 0 , where p is an odd prime and m is any positive integer. This article delves into the algebraic structure of skew negacyclic codes of length 2p^s over a finite field \mathbb{F}_{p^m} and a finite chain ring \mathfrak{R} . The focus is on the classification and structural properties of these codes. Based on the different possible factorizations of x^{2p^s} + 1 over \mathbb{F}_{p^m} , a complete classification of the structural properties of skew negacyclic codes and their duals for length 2p^s over \mathbb{F}_{p^m} and \mathfrak{R} is provided. Furthermore, the algebraic structure of \mathbb{F}_{p^m}\mathfrak{R} -additive skew negacyclic codes with block length (p^s, 2p^s) is discussed. The separability of \mathbb{F}_{p^m}\mathfrak{R} -additive skew negacyclic codes is also analyzed. To illustrate these results, several examples are presented, including the construction of Maximum Distance Separable (MDS) and near-MDS codes.