The linear programming optimum for packings in classical association schemes

TL;DR

This paper determines the optimal solutions of linear programs related to $d$-codes in most classical association schemes, leading to new bounds on intersecting sets and extending combinatorial theorems.

Contribution

It provides the first complete solutions to the linear programming problem for nearly all classical association schemes, excluding Hamming and Johnson schemes.

Findings

Optimal linear program solutions for most classical schemes

New upper bounds on $t$-intersecting sets in association schemes

Improved bounds on $t$-intersecting sets of generators in polar spaces

Abstract

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 Erd\H{o}s-Ko-Rado Theorem in extremal set theory. Our proofs draw on techniques from algebraic combinatorics and the duality theory of linear programming.