A geometric approach to generalized covering radii of linear codes

2026-06-15

AI Summary

This work investigates the generalized covering radius of linear codes from a geometric perspective, introducing for the first time the notion of $(\rho,t)$-saturating sets as their counterpart in finite geometry, thereby unifying classical saturating sets and $t$-strong blocking sets. By integrating tools from finite geometry, the dual Grassmannian criterion, affine methods, and combinatorial configuration techniques, the paper establishes novel connections between coding theory and finite geometry. The main contributions include multiple equivalent characterizations and lower bounds on the size of $(\rho,t)$-saturating sets, along with efficient construction methods derived from strong blocking sets, graphs, and projective configurations. These results provide a systematic framework for the analysis and application of generalized covering radii in coding theory.

Abstract

Covering problems in coding theory are closely related to finite geometry through the interpretation of the columns of parity-check matrices as point sets in finite vector spaces. Motivated by the recent notion of generalized covering radii of linear codes introduced by Elimelech, Firer and Schwartz, we develop a geometric framework for these parameters. We introduce $(ρ,t)$-saturating sets and show that they are precisely the finite-geometric counterparts of linear codes whose $t$-th generalized covering radius is at most $ρ$. We study the structure of these sets and show that the extremal case $ρ=t$ coincides with the notion of $t$-strong blocking sets. Thus, $(ρ,t)$-saturating sets interpolate between classical saturating sets and strong blocking sets. We provide several equivalent formulations, including affine and dual Grassmannian criteria, derive lower bounds on their size, and give constructions from strong blocking sets, graphs and projective configurations.

Problem

• generalized covering radii
• linear codes
• finite geometry
• saturating sets
• blocking sets

Innovation

• generalized covering radius
• (ρ,t)-saturating sets
• finite geometry
• strong blocking sets
• linear codes