Unique Optima of the Delsarte Linear Program

TL;DR

This paper investigates the conditions under which the Delsarte linear program has a unique optimal solution, revealing parity phenomena and symmetries in the solutions related to code extension and puncturing.

Contribution

It establishes when the Delsarte linear program has a unique optimum and introduces the Krawtchouk decomposition to analyze solution structures, uncovering parity and symmetry properties.

Findings

Unique optima occur if d > n/2 or d ≤ 2.

Existence of optima with identical Krawtchouk decompositions for certain parameters.

Parity and symmetry properties reduce decision variables in the linear program.

Abstract

The Delsarte linear program is used to bound the size of codes given their block length $n$ and minimal distance $d$ by taking a linear relaxation from codes to quasicodes. We study for which values of $(n,d)$ this linear program has a unique optimum: while we show that it does not always have a unique optimum, we prove that it does if $d>n/2$ or if $d \leq 2$. Introducing the Krawtchouk decomposition of a quasicode, we prove there exist optima to the $(n,2e)$ and $(n-1,2e-1)$ linear programs that have essentially identical Krawtchouk decompositions, revealing a parity phenomenon among the Delsarte linear programs. We generalize the notion of extending and puncturing codes to quasicodes, from which we see that this parity relationship is given by extending/puncturing. We further characterize these pairs of optima, in particular demonstrating that they exhibit a symmetry property, effectively halving the number of decision variables.