  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.
