Unique Optima of the Delsarte Linear Program
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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 ≤ 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.
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