Skew cyclic codes of arbitrary length

Abstract

In this paper we study a special type of linear codes, called skew cyclic codes, in the most general case. This set of codes is a generalization of cyclic codes but constructed using a non-commutative ring called the skew polynomial ring. In previous works these codes have been studied with certain restrictions on their length. This work examines their structure for an arbitrary length without any restriction. Our results show that these codes are equivalent to either cyclic 1 codes or quasi-cyclic codes, hence establish strong connections with well-known classes of codes.

A Note on the Existence of Self-Dual Skew Codes over Finite Fields

“…According to Theorem 18 of [14], a θ-cyclic code of length n is equivalent to a quasi-cyclic code of index where = gcd(|θ|, n). Therefore, as equivalence preserves self-duality, if there exists a self-dual θ-cyclic code of length n then there exists a self-dual quasi-cyclic code of length n and index = gcd(|θ|, n).…”

“…The following Lemma is inspired from Theorem 16 and Theorem 18 of [14] which state that a θ-cyclic code is either a cyclic code or a quasi-cyclic code.…”

Skew quasi-cyclic codes over Galois rings

“…A principal module σ -cyclic code is one of the form Rg(x)/R(x n − 1) for some g(x) ∈ R. Over finite fields, module σ -codes have been investigated extensively by Boucher and Ulmer [7]. Recently, Siap et al [18] have also studied skew cyclic codes of arbitrary lengths.…”

On the Codes over a Semilocal Finite Ring

“…Some authors generalized the notion of cyclic, quasi-cyclic and constacyclic codes by using generator polynomials in skew polynomial rings [1,2,5,7,8,9,14,15,18,27,30].…”