Combinatorial Bounds for Codes over Metric Spaces: Ramsey-Sidorenko Thresholds and Subgraph Counts

2026-07-29

AI Summary

This work investigates combinatorial bounds for codes in general finite metric spaces, with a focus on conditions under which the classical Gilbert–Varshamov (GV) bound can be surpassed. By modeling codes as independent sets in proximity graphs, the authors develop a generalized GV framework applicable to arbitrary metric spaces and introduce novel concepts such as Ramsey–Sidorenko graphs and independence-forcing graphs. Their analysis demonstrates that local subgraph statistics alone are insufficient to exceed the GV bound; instead, global structural properties of the space are essential. Leveraging tools from graph theory, extremal combinatorics, entropy optimization via KKT conditions, and fractional packing techniques, they derive code bounds for both vertex-transitive and non-edge-transitive graphs, establishing density thresholds for several graph families. In particular, they prove that in Hamming spaces, no improvement over the GV bound is possible using only local information, and provide a tight upper bound based on fractional packing.

Abstract

This paper investigates the relationship between coding theory and extremal combinatorics by representing codes in general metric spaces as independent sets in proximity graphs. We provide a generalized framework for the Gilbert-Varshamov (GV) bound applicable to codes over any finite metric space and explore the conditions under which global combinatorial parameters can force the existence of codes exceeding this bound. Central to our analysis is the introduction of Ramsey-Sidorenko and independence-forcing graphs. We establish density thresholds for various graph families and utilize the Karush--Kuhn--Tucker conditions to analyze entropy optimization in the Hamming case. Furthermore, we derive upper bounds on code sizes using fractional packings in vertex-transitive and nonedge-transitive graphs. Our findings demonstrate that local subgraph statistics alone are insufficient to surpass the GV bound in the Hamming case, suggesting that improvements must stem from large-scale structural properties of the space.

Problem

• coding theory
• Gilbert-Varshamov bound
• extremal combinatorics
• metric spaces
• Ramsey-Sidorenko

Innovation

• Ramsey-Sidorenko graphs
• Gilbert-Varshamov bound
• fractional packings
• independence-forcing graphs
• entropy optimization