Skew cyclic codes over \(\mathbb{F}_{p} + u\mathbb{F}_{p}\) (Q1993391)

Summary: 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 \(u^{2} = 0\). We characterise all skew cyclic codes of length \(n\) as left \(R[x;\theta]\)-submodules of \(R_{n} = R[x;\theta] / \langle x^{n} - 1\rangle\). We find all generator polynomials for these codes and describe their minimal spanning sets. Moreover, an encoding 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 primes \(p\): In fact, example 6 in our paper is a new ternary code in the class of quasi-twisted codes. We also present several examples of optimal codes.