Unique optima of the Delsarte linear program
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10623_2023_Article_1191.pdf
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378.09 KB
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Author(s)
Li, Rupert
Date Issued
January 29, 2023
Publisher
Springer US
Citation
Li, Rupert. 2023. "Unique optima of the Delsarte linear program."
Version
Final published version
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1007/s10623-023-01191-y
