On defining generalized rank weights

Abstract

This paper investigates the generalized rank weights, with a definition implied by the study of the generalized rank weight enumerator. We study rank metric codes over $L$, where $L$ is a finite Galois extension of a field $K$. This is a generalization of the case where $K = \mathbb{F}_q$ and $L = \mathbb{F}_{q^m}$ of Gabidulin codes to arbitrary characteristic. We show equivalence to previous definitions, in particular the ones by Kurihara-Matsumoto-Uyematsu, Oggier-Sboui and Ducoat. As an application of the notion of generalized rank weights, we discuss codes that are degenerate with respect to the rank metric.Comment: 15 pages; extended abstract accepted for presentation at ACA2015 (http://www.usthb.dz/spip.php?article1039

Rank-metric codes and q-polymatroids

“…We adopt the definition from [17] and, in Section 2, we show that it is invariant with respect to equivalence of rank-metric codes. We also show that the definition of generalized weights for rank-metric codes from [10], which generalizes definitions from [11,24,8], is not invariant with respect to code equivalence.…”

“…Notice that one could also define generalized weights for rank-metric codes following a support-based analogy with codes endowed with the Hamming metric. This naturally leads to generalizing the invariants proposed in [11], [24] and [8] as in the following Definition 2.6. This approach has been followed, e.g., in [10].…”

“…Two of the first definitions of generalized weights for vector rankmetric codes are due to Kurihara, Matsumoto, and Uyematsu [12] and to Oggier and Sboui [13]. More definitions are due to Jurrius and Pellikaan [8] and Martínez-Peñas and Matsumoto [10], who also compared the various definitions. Using the theory of anticodes, Ravagnani [17] gave a definition of generalized weights for matrix rank-metric codes, which extends the one from [12].…”

A geometric approach to rank metric codes and a classification of constant weight codes

“…There are several approaches for the notion of generalized weights for rank metric codes [JP17, KMU15, OS12, DK15, Rav16]. These existing definitions were shown to be equivalent in [JP17].…”

Theory of supports for linear codes endowed with the sum-rank metric

“…These characterizations naturally recover well-known characterizations of Hammingmetric and rank-metric support spaces. Item 2 in the case n = 1 is the well-known fact that Hamming-support spaces are cartesian products of factors {0} or F. Item 3 in the case ℓ = 1 corresponds to the characterization in [24,Th. 4.3 & Prop.…”