On self-dual skew cyclic codes of length \(p^s\) over \(\mathbb{F}_{p^m} + u \mathbb{F}_{p^m} \)
For a finite field \(\mathbb{F}_{q},\) and \(\mathcal{R}_2 = {\mathbb{F}}_{p^m} + u{\mathbb{F}}_{p^m},\) where \(u^2 = 0,\) a skew-cyclic code is a left ideal of the non-commutative ring \(\mathbb{F}_{q}[x, \theta],\) where and \({\theta}\) is an automorphism of \(\mathbb{F}_{q}.\) Skew cyclic codes of length \(p^s\) over \(\mathcal{R}_2\) are divided into four types: Type 1 (trivial ideals), Type 2 (principal left ideals with non-monic polynomial generators), Type 3 (principal left ideals with monic polynomial generators) and Type 4 (non-principal left ideals). Any polynomial \(f(x)\in\mathcal{R}_2\) can be uniquely expressed as \(f(x)=\sum\limits_{i=0}^{p^s-1}{a_{0i}(x-1)^i}+u\sum\limits_{i=0}^{p^s-1}{a_{1i}(x-1)^i},\) where \(a_{0i}, a_{1i}\in\mathbb{F}_{p^m}.\) The main theme of this work are the special skew cyclic codes such that \(a(x),a_1(x),a_2(x)\in\{(x-1)^j : 0\leq j\leq ps-1\}.\) For \(a(x)\) -- a monic factor of \(x^{p^s}-1\) the sum \((x-1)^na\) is calculated for a skew cyclic code of Type 3 (principal left ideals with monic polynomial generators) \(C =\mathcal{R}_2((x-1)^i +uh(x)(x-1)^t)\) and some additional properties are also shown. For all four types, the number of codewords \(n_C\) of \(C\) is determined as well as the dual of these codes and the self-dual among them are obtained. Some examples of self-dual skew cyclic codes of length \(p^s\) over \(\mathcal{R}_2\) are shown.
scientific article; zbMATH DE number 7391497
scientific article