Semidefinite programming bounds for binary codes from a split Terwilliger algebra

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 $6551$.