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 $6551$.

翻译：我们研究$A(n,d)$的上界，即长度为$n$且汉明距离至少为$d$的码字的最大尺寸。Schrijver研究了汉明方案的Terwilliger代数，并提出了一种用于界定$A(n,d)$的半定规划方法。我们基于分裂Terwilliger代数推导出更精细的矩阵不等式，从而改进了Schrijver关于$A(n,d)$的半定规划界。特别地，我们将$A(18,4)$的半定规划界改进至$6551$。