The linear programming optimum for packings in classical association schemes

2025-08-18

AI Summary

This work addresses the linear programming (LP) upper bound problem for the maximum size of $d$-codes in classical association schemes beyond the Hamming and Johnson schemes. Method: We construct feasible primal and dual LP solutions and rigorously prove their objective values coincide, thereby determining the exact LP optimum for nearly all non-classical cases. Our approach integrates distance-regular graph theory, algebraic combinatorics, and strong LP duality techniques to systematically analyze code structures and intersection properties. Contribution/Results: The results yield new tight LP upper bounds on $d$-codes in several classical schemes—including $q$-analogs—and improve existing bounds on $t$-intersecting families in extremal vector spaces. Moreover, our framework unifies and reproves Erdős–Ko–Rado-type theorems across multiple settings, significantly advancing the frontiers of algebraic coding theory and extremal combinatorics.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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. This is achieved by constructing feasible solutions to both the primal and dual linear programs whose objective values coincide. 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.

Problem

• Determines linear programming optimum for packings in classical association schemes
• Studies d-codes in Hamming, Johnson, and q-analog schemes
• Improves bounds on t-intersecting sets in classical association schemes

Innovation

• Linear programming optimizes classical association schemes
• Dual solutions match primal for exact bounds
• New upper bounds for t-intersecting sets