Skew cyclic codes over ${\mathbb{F}}_p+u{\mathbb{F}}_p$

Abstract

In this paper, we study skew cyclic codes with arbitrary length over the ring $R={\mathbb{F}}_p+u{\mathbb{F}}_p$ where $p$ is an odd prime and . We characterize all skew cyclic codes of length $n$ as left -submodules of $R_n=R[x;\theta ]/⟨x^n-1⟩$. We find all generator polynomials for these codes and describe their minimal spanning sets. Moreover, an encoding and decoding algorithm is presented for skew cyclic codes over the ring $R$. Finally, based on the theory we developed in this paper, we provide examples of codes with good parameters over $F_p$ with different odd prime $p.$ In fact, example 25 in our paper is a new ternary code in the class of quasi-twisted codes. The other examples we provided are examples of optimal codes.