Improved Linear Programming Bounds on Sizes of Constant-Weight Codes

TL;DR

This paper enhances linear programming bounds for constant-weight binary codes, providing new upper bounds for code sizes up to length 28 and simplifying key proofs in the theory.

Contribution

It introduces additional constraints to Delsarte's linear programming method, resulting in twenty-three new upper bounds and a simplified proof of a fundamental theorem.

Findings

Twenty-three new upper bounds for A(n,d,w) for n ≤ 28

Simplified proof of Delsarte's theorem for binary codes

Enhanced linear programming techniques for code size estimation

Abstract

Let $A(n,d,w)$ be the largest possible size of an $(n,d,w)$ constant-weight binary code. By adding new constraints to Delsarte linear programming, we obtain twenty three new upper bounds on $A(n,d,w)$ for $n \leq 28$. The used techniques allow us to give a simple proof of an important theorem of Delsarte which makes linear programming possible for binary codes.