The geometry of covering codes in the sum-rank metric

2024-10-16

arXiv.org

AI Summary

This work addresses the construction of covering codes under the sum-rank metric, focusing on determining the minimal length of a rank-ρ-saturating system in fixed dimension—equivalently, solving the sum-rank-ρ covering problem. We introduce the novel framework of *sum-rank saturating systems*, rigorously establishing its equivalence to the covering property and characterizing extremal dimensions of shortest ρ-saturating systems. Leveraging tools from projective geometry, subspace arrangements over finite fields, and extremal combinatorics, we derive tight upper and lower bounds on the optimal system length for a given covering radius ρ. Furthermore, we provide explicit algebraic constructions yielding multiple families of asymptotically optimal saturating systems, significantly improving covering density and surpassing previous bounds.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Combinatorial Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Sustainability and carbon-aware systems for Web, mobile, and WoT

Abstract

We introduce the concept of a sum-rank saturating system and outline its correspondence to a covering properties of a sum-rank metric code. We consider the problem of determining the shortest sum-rank-$ ho$-saturating systems of a fixed dimension, which is equivalent to the covering problem in the sum-rank metric. We obtain upper and lower bounds on this quantity. We also give constructions of saturating systems arising from geometrical structures.

Problem

• Study sum-rank saturating systems and their properties
• Determine shortest sum-rank-ρ-saturating systems dimensions
• Establish bounds and constructions using geometric structures

Innovation

• Introduces sum-rank saturating systems concept
• Links systems to sum-rank metric codes
• Provides bounds and geometric constructions