One-weight codes in the sum-rank metric

2025-08-06

AI Summary

This paper investigates the classification and geometric structure of one-weight linear codes under the sum-rank metric—codes where all nonzero codewords share the same sum-rank weight. Employing finite geometry, linear algebra, and combinatorial analysis, we achieve the first complete classification of constant rank-list sum-rank codes. We construct the first explicit examples of constant rank-profile codes and provide a structural characterization thereof. Furthermore, we uncover the geometric essence of one-weight maximum sum-rank distance (MSRD) codes: in dimension two, their existence is equivalent to partitions of discrete linear sets on the projective line; in dimension three, it corresponds to 2-fold blocking sets in the projective plane. These insights yield novel existence bounds and nonexistence results, establishing a systematic geometric framework for extremal codes under the sum-rank metric.

Technology Category

• Machine Learning: Learning Preferences or Rankings
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: Remote Sensing / Geospatial AI

Application Category

• Search and Retrieval-Augmented AI: Web search models and ranking
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Large-scale security measurements

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.

Problem

• Classify one-weight sum-rank metric codes with constant rank-list
• Investigate constant rank-profile sum-rank codes and provide partial results
• Explore one-weight MSRD codes and their geometric connections

Innovation

• Classify constant rank-list sum-rank codes
• Explore constant rank-profile codes structure
• Link MSRD codes to projective geometry