New Solutions to Delsarte's Dual Linear Programs

TL;DR

This paper develops new universal bounds for codes within association schemes by constructing novel solutions to Delsarte's dual linear programs, extending classical bounds and providing a unified framework applicable to various code types.

Contribution

It introduces generalized bounds for association schemes and constructs new solutions to Delsarte's dual LP, inspired by Laplacian methods, broadening the scope of coding theory bounds.

Findings

Recovered known bounds for q-ary and constant-weight binary codes.

Constructed new solutions to Delsarte's dual LP applicable to general association schemes.

Provided a Laplacian-inspired approach to the second linear programming bound.

Abstract

Understanding the maximum size of a code with a given minimum distance is a major question in computer science and discrete mathematics. The most fruitful approach for finding asymptotic bounds on such codes is by using Delsarte's theory of association schemes. With this approach, Delsarte constructs a linear program such that its maximum value is an upper bound on the maximum size of a code with a given minimum distance. Bounding this value can be done by finding solutions to the corresponding dual linear program. Delsarte's theory is very general and goes way beyond binary codes. In this work, we provide universal bounds in the framework of association schemes that generalize the Elias-Bassalygo bound, which can be applied to any association scheme constructed from a distance function. These bounds are obtained by constructing new solutions to Delsarte's dual linear program. We instantiate these results and we recover known bounds for $q$-ary codes and for constant-weight binary codes. Our other contribution is to recover, for essentially any $Q$-polynomial scheme, MRRW-type solutions to Delsarte's dual linear program which are inspired by the Laplacian approach of Friedman and Tillich instead of using the Christoffel-Darboux formulas. We show in particular how the second linear programming bound can be interpreted in this framework.