New Solutions to Delsarte's Dual Linear Programs
André Chailloux, Thomas Debris-Alazard
TL;DR
The paper advances the theory of Delsarte-type bounds by presenting universal dual linear-programming solutions within the framework of distance-induced association schemes, enabling generalized Elias–Bassalygo bounds that apply to any $P$-polynomial scheme. It further extends the classical MR_RW bounds through a dual-Laplacian approach that leverages $Q$-polynomial structure to produce MR_RW-type bounds for essentially any such scheme, including a second-LP-bound interpretation. The results recover known bounds for $q$-ary and constant-weight binary codes and unify LP-based bounds with Laplacian techniques, offering a versatile toolkit for asymptotic rate bounds. Applications to the Hypercube and Johnson schemes demonstrate concrete instantiations, yielding explicit asymptotics and improving understanding of the trade-offs between distance, weight, and alphabet size in coding theory.
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.