Semidefinite programming bounds for binary codes from a split Terwilliger algebra

TL;DR

This paper improves upper bounds on the maximum size of binary codes with given length and minimum Hamming distance by developing advanced semidefinite programming techniques using a split Terwilliger algebra.

Contribution

It introduces a more sophisticated matrix inequality framework based on split Terwilliger algebra to enhance existing semidefinite bounds for binary codes.

Findings

Improved bound on A(18,4) to 6551

Enhanced semidefinite programming bounds for binary codes

New matrix inequalities based on split Terwilliger algebra

Abstract

We study the upper bounds for $A(n,d)$, the maximum size of codewords with length $n$ and Hamming distance at least $d$. Schrijver studied the Terwilliger algebra of the Hamming scheme and proposed a semidefinite program to bound $A(n, d)$. We derive more sophisticated matrix inequalities based on a split Terwilliger algebra to improve Schrijver's semidefinite programming bounds on $A(n, d)$. In particular, we improve the semidefinite programming bounds on $A(18,4)$ to $6551$.