On Nearly-Perfect Covering Codes Beyond Radius One

2026-08-12

AI Summary

This study addresses the existence of nearly perfect covering codes attaining equality in the Van Wee bound for covering radii \( R \geq 2 \). By incorporating the minimum distance of a code into the Van Wee bound, the work establishes for the first time that the original bound is unattainable when \( R \geq 2 \), and introduces a strictly tighter improved bound. Leveraging tools from algebraic coding theory and combinatorial analysis, the authors completely classify all equivalence classes of nearly perfect covering codes for \( R = 2 \) and \( R = 3 \), and further prove that for any fixed \( R \geq 3 \), only finitely many such codes exist. This work thus provides a systematic characterization of the structural limitations and existence boundaries of nearly perfect covering codes in the high-radius regime.

Abstract

We study (binary) nearly-perfect covering codes, which are codes that attain the Van Wee bound with equality. They act as the covering counterparts to nearly-perfect error-correcting codes, which attain the Johnson bound with equality. These codes have been completely classified for covering radius $R=1$. We prove that no code with $R\geq 2$ can attain the original Van Wee bound with equality, since it omits the dependence on the minimum distance of the code. We refine the bound to account for the minimum distance and show some nearly-perfect covering codes. By proving some structural properties of such codes, we prove all nearly-perfect covering codes with $R=2,3$ must be equivalent to the codes we showed. We also prove that for any $R\geq 3$, there are at most a finite number of nearly-perfect covering codes.

Problem

• covering codes
• nearly-perfect codes
• covering radius
• Van Wee bound
• minimum distance

Innovation

• nearly-perfect covering codes
• Van Wee bound
• covering radius
• minimum distance
• code classification