Two-weight rank-metric codes

TL;DR

The paper addresses the problem of classifying two-weight linear rank-metric codes. It develops a geometric framework based on ${ m mathbb{F}}_q$-linear sets and ${ m mathbb{F}}_{q^e}$-scattered subspaces to connect codes with dual linear sets and ${ m q}$-systems, establishing a complete characterization: every nondegenerate two-weight $[n,k,d]_{q^m/q}$ code is the dual of an ${ m mathbb{F}}_{q^{m-d}}$-scattered subspace, yielding nonzero weights $d$ and $m$. In the antipodal case, for $q\ge m$, such codes come from ${ m mathbb{F}}_{q^{n-d}}$-scattered subspaces and are induced by MRD codes over an extension; the work also provides existence conditions and length bounds under divisibility constraints and analyzes puncturing, linking the two-weight property to maximal scattering. By tying two-weight rank-metric codes to scattered subspaces, the results enable constructive existence and dimension bounds, advancing both theory and potential applications in network coding and combinatorial design.

Abstract

Two-weight linear codes are linear codes in which any nonzero codeword can have only two possible distinct weights. Those in the Hamming metric have proven to be very interesting for their connections with authentication codes, association schemes, strongly regular graphs, and secret sharing schemes. In this paper, we characterize two-weight codes in the rank metric, answering a recent question posed by Pratihar and Randrianarisoa.

Theorems & Definitions (30)

• Theorem 1.1
• Theorem 2.1
• Theorem 2.2
• Proposition 2.3
• Definition 2.4
• Definition 2.5
• Theorem 2.6
• Definition 2.7
• Theorem 3.1
• Theorem 3.2