Hermitian Rank Metric Codes and Duality

Abstract

In this paper we define and study rank metric codes endowed with a Hermitian form. We analyze the duality for F q 2-linear matrix codes in the ambient space (F q 2) n,m and for both F q 2-additive codes and F q 2m-linear codes in the ambient space F n q 2m. Similarly, as in the Euclidean case we establish a relationship between the duality of these families of codes. For this we introduce the concept of q mduality between bases of F q 2m over F q 2 and prove that a q m-self dual basis exists if and only if m is an odd integer. We obtain connections on the dual codes in F n q 2m and (F q 2) n,m with the corresponding inner products. In particular, we study Hermitian linear complementary dual, Hermitian self-dual and Hermitian self-orthogonal codes in F n q 2m and (F q 2) n,m. Furthermore, we present connections between Hermitian F q 2additive codes and Euclidean F q 2-additive codes in F n q 2m. INDEX TERMS Rank metric codes-Additive rank metric codes-Hermitian rank metric codes This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.

Rank Metric Codes and Their Galois Duality

“…From the similar results established in [19] with respect to dual bases, and in [4] with respect to Hermitian inner product and q m -dual bases. Now for F Q m -linear rank metric code C ⊆ F n Q m and our given q sm -dual bases, we can obtain the result that λ B (C ⊥s ) = λ B (C) ⊥ M s as follows.…”

“…In recent years, the duality of rank metric codes has been extensively researched( [12], [14], [17], [19]). Cruz et al [4] defined the Hermitian form and introduced F q 2linear, F q 2 -additive and F q 2m -linear rank metric codes. The connections of duality with corresponding Hermitian inner products of those codes and the existence of q m -self-dual basis with respect to Hermitian form has been studied.…”

Encoding and Decoding of Several Optimal Rank Metric Codes

“…We choose interpolation-based decoding approach which is able to decode errors up to half of the minimum distance in polynomial time for all the aforementioned codes. A part of our work in this paper responds to a suggestion in [1], where the authors suggested studying the decoding of Hermitian rank metric codes.…”

Galois Hull Dimensions of Gabidulin Codes

“…A special class of maximum distance separable (MDS) codes, called Gabidulin codes was introduced in [13]. Detailed surveys of Gabidulin codes and their (Euclidean/Hermitian) duality can be found in [6], [7], [9], [19], [30]. Here, we investigate the e-Galois hull dimension of Gabidulin codes, in particular, by using a self-dual basis of F q m over F q , we H. Islam and A-L. Horlemann are with School of Computer Science, University of St. Gallen, Switzerland, (e-mail: habibul.islam@unisg.ch, annalena.horlemann@unisg.ch).…”