{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T07:21:17Z","timestamp":1740122477903,"version":"3.37.3"},"reference-count":12,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2023,1,29]],"date-time":"2023-01-29T00:00:00Z","timestamp":1674950400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,1,29]],"date-time":"2023-01-29T00:00:00Z","timestamp":1674950400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Des. Codes Cryptogr."],"published-print":{"date-parts":[[2023,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The Delsarte linear program is used to bound the size of codes given their block length <jats:italic>n<\/jats:italic> and minimal distance <jats:italic>d<\/jats:italic> by taking a linear relaxation from codes to quasicodes. We study for which values of (<jats:italic>n<\/jats:italic>,\u00a0<jats:italic>d<\/jats:italic>) 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 <jats:inline-formula><jats:alternatives><jats:tex-math>$$d&gt;n\/2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>d<\/mml:mi>\n                    <mml:mo>&gt;<\/mml:mo>\n                    <mml:mi>n<\/mml:mi>\n                    <mml:mo>\/<\/mml:mo>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> or if <jats:inline-formula><jats:alternatives><jats:tex-math>$$d \\le 2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>d<\/mml:mi>\n                    <mml:mo>\u2264<\/mml:mo>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Introducing the Krawtchouk decomposition of a quasicode, we prove there exist optima to the (<jats:italic>n<\/jats:italic>,\u00a02<jats:italic>e<\/jats:italic>) and <jats:inline-formula><jats:alternatives><jats:tex-math>$$(n-1,2e-1)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>n<\/mml:mi>\n                    <mml:mo>-<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mn>2<\/mml:mn>\n                    <mml:mi>e<\/mml:mi>\n                    <mml:mo>-<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> 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.<\/jats:p>","DOI":"10.1007\/s10623-023-01191-y","type":"journal-article","created":{"date-parts":[[2023,1,29]],"date-time":"2023-01-29T10:02:57Z","timestamp":1674986577000},"page":"2117-2140","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Unique optima of the Delsarte linear program"],"prefix":"10.1007","volume":"91","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6870-087X","authenticated-orcid":false,"given":"Rupert","family":"Li","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,1,29]]},"reference":[{"issue":"1","key":"1191_CR1","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1016\/0021-9045(90)90072-X","volume":"60","author":"L Chihara","year":"1990","unstructured":"Chihara L., Stanton D.: Zeros of generalized Krawtchouk polynomials. 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