One-weight codes in the sum-rank metric
Usman Mushrraf, Ferdinando Zullo
TL;DR
This work advances the geometry of one-weight codes in the sum-rank metric by developing a geometric framework based on linear sets to study three main directions: constant rank-list codes, constant rank-profile codes, and one-weight MSRD codes. It proves a complete classification for constant rank-list sum-rank codes, showing that projections are simplex rank-metric codes and constraining parameters accordingly, while providing initial constructions and structural results for constant rank-profile codes. For MSRD codes, it shows dimension-two constructions arise from partitions of scattered linear sets on projective lines and connects dimension-three existence to 2-fold blocking sets in the projective plane, yielding new bounds and nonexistence results over certain fields. The results collectively deepen the link between sum-rank code geometry and finite projective-space configurations, with implications for optimality (MSRD) and code design in layered metric settings.
Abstract
One-weight codes, in which all nonzero codewords share the same weight, form a highly structured class of linear codes with deep connections to finite geometry. While their classification is well understood in the Hamming and rank metrics - being equivalent to (direct sums of) simplex codes - the sum-rank metric presents a far more intricate landscape. In this work, we explore the geometry of one-weight sum-rank metric codes, focusing on three distinct classes. First, we introduce and classify \emph{constant rank-list} sum-rank codes, where each nonzero codeword has the same tuple of ranks, extending results from the rank-metric setting. Next, we investigate the more general \emph{constant rank-profile} codes, where, up to reordering, each nonzero codeword has the same tuple of ranks. Although a complete classification remains elusive, we present the first examples and partial structural results for this class. Finally, we consider one-weight codes that are also MSRD (Maximum Sum-Rank Distance) codes. For dimension two, constructions arise from partitions of scattered linear sets on projective lines. For dimension three, we connect their existence to that of special $2$-fold blocking sets in the projective plane, leading to new bounds and nonexistence results over certain fields.