On the roots and minimum rank distance of skew cyclic codes

Abstract

Skew cyclic codes play the same role as cyclic codes in the theory of error-correcting codes for the rank metric. In this paper, we give descriptions of these codes by root spaces, cyclotomic spaces and idempotent generators. We prove that the lattice of skew cyclic codes is anti-isomorphic to the lattice of root spaces, study these two lattices and extend the rank-BCH bound on their minimum rank distance to rank-metric versions of the van Lint-Wilson's shift and Hartmann-Tzeng bounds. Finally, we study skew cyclic codes which are linear over the base field, proving that these codes include all Hamming-metric cyclic codes, giving then a new relation between these codes and rank-metric skew cyclic codes

Hartmann–Tzeng bound and skew cyclic codes of designed Hamming distance

“…Recently, a HT bound with respect to a rank metric has been obtained in [27,Corollary 5], in the realm of skew block codes. Setting L = F q m in Theorem 3.3 we get a similar statement to [27,Corollary 5]. The precise relation between both results is explained in Proposition A.5 and Remark A.6 in the Appendix.…”

“…It is pertinent to compare Theorem 3.3, when applied to finite fields, with [27,Corollary 5]. In order to work with exactly the same general hypotheses than in [27], we note that Section 3 is still valid, word by word, without requiring that L σ = M θ as in Definition 2.2. To be precise, let M be a finite field extension of L, and θ : M → M a field automorphism of finite order n such that, by restriction, gives an automorphism σ : L → L. Set R = L[x; σ] and S = M [x; θ].…”

“…If we set K = M θ , then all statements and proofs of Section 3 are still valid. Now, let us briefly recall the scenario where [27] is represented. Let k, m, n positive integers such that m divides kn, so that F q m is considered as a subfield of F q kn .…”

“…This kind of roots of x n − 1 is used in Sections 3 and 4 to give a Hartmann-Tzeng-like bound of the Hamming distance of skew cyclic codes and to build codes with designed distance. Theorem 3.3 may be seen as an extension of [27,Corollary 5] to a general context, which includes skew convolutional codes, and, similarly, Corollary 3.4 can be considered as an extension of [9, Proposition 1]. However, our approach and methods are conceptually different from that of [27], based upon root spaces, or from the Picard-Vessiot view used in [9].…”

“…Theorem 3.3 may be seen as an extension of [27,Corollary 5] to a general context, which includes skew convolutional codes, and, similarly, Corollary 3.4 can be considered as an extension of [9, Proposition 1]. However, our approach and methods are conceptually different from that of [27], based upon root spaces, or from the Picard-Vessiot view used in [9]. Our method of constructing skew codes of prescribed parameters is based on a very simple combinatorial procedure, namely, compute the closure of suitable subsets of {0, 1, .…”