Unique optima of the Delsarte linear program

Abstract

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

$$d>n/2$$

d > n / 2

or if

$$d \le 2$$

d ≤ 2

. Introducing the Krawtchouk decomposition of a quasicode, we prove there exist optima to the (n, 2e) and

$$(n-1,2e-1)$$

( n - 1 , 2 e - 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.