Association schemes are central objects in algebraic combinatorics, with the classical schemes lying at their core. These classical association schemes essentially consist of the Hamming and Johnson schemes, and their $q$-analogs: bilinear forms scheme, alternating bilinear forms scheme, Hermitian forms scheme, $q$-Johnson scheme, and polar space schemes. Each of them gives rise to a distance-regular graph on a vertex set $X$, naturally endowed with the path metric. We study $d$-codes in these schemes, that is, subsets $Y$ of $X$ in which every pair of distinct elements has path distance at least $d$. A powerful tool for deriving upper bounds on the size of $d$-codes is the linear programming method. In the case of the Hamming and Johnson schemes, the linear program has been studied since the 1970s, but its optimum is still unknown. We determine the optimum of the linear program for nearly all classical association schemes distinct from the Hamming and Johnson schemes. As a corollary, we obtain upper bounds on $t$-intersecting sets in classical association schemes, providing new proofs of several known results and, in particular, improving earlier bounds on $t$-intersecting sets of generators in polar spaces. These results can be viewed as analogs of the classical Erdős-Ko-Rado Theorem in extremal set theory. Our proofs draw on techniques from algebraic combinatorics and the duality theory of linear programming.

翻译：结合方案是代数组合学中的核心对象，经典结合方案更处于其中心地位。这些经典结合方案主要包括Hamming方案、Johnson方案及其$q$-模拟：双线性形式方案、交错双线性形式方案、Hermite形式方案、$q$-Johnson方案与极空间方案。每个方案都在顶点集$X$上诱导出一个距离正则图，自然配备路径度量。我们研究这些方案中的$d$-码，即$X$的子集$Y$，其中任意两个不同元素之间的路径距离至少为$d$。推导$d$-码大小的上界时，线性规划方法是一个强大工具。对于Hamming方案和Johnson方案，线性规划自20世纪70年代起便得到研究，但其最优解至今未知。我们确定了除Hamming方案和Johnson方案外几乎所有经典结合方案中线性规划的最优解。作为推论，我们得到了经典结合方案中$t$-相交集的上界，为若干已知结果提供了新证明，并特别改进了极空间中生成元$t$-相交集的先前上界。这些结果可视为极值集合论中经典Erdős-Ko-Rado定理的类似物。我们的证明运用了代数组合学与线性规划对偶理论中的方法。