# Semidefinite programming bounds for constant weight codes

## TL;DR

This paper develops semidefinite programming bounds to improve the upper limits of constant weight codes, providing exact values for specific parameters and insights into the structure of Golay codes.

## Contribution

It introduces two new semidefinite programming bounds for constant weight codes, leading to exact determinations of $A(22,8,10)$ and $A(22,8,11)$, and analyzes Golay code structures.

## Key findings

- New upper bounds for $A(22,8,10)$ and $A(22,8,11)$
- Exact values for specific constant weight codes
- Golay code structure as a union of constant weight codes

## Abstract

For nonnegative integers $n,d,w$, let $A(n,d,w)$ be the maximum size of a code $C \subseteq \mathbb{F}_2^n$ with constant weight $w$ and minimum distance at least $d$. We consider two semidefinite programs based on quadruples of code words that yield several new upper bounds on $A(n,d,w)$. The new upper bounds imply that $A(22,8,10)=616$ and $A(22,8,11)=672$. Lower bounds on $A(22,8,10)$ and $A(22,8,11)$ are obtained from the $(n,d)=(22,7)$ shortened Golay code of size $2048$. It can be concluded that the shortened Golay code is a union of constant weight $w$ codes of sizes $A(22,8,w)$.