2026-06-02T16:34:24Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:17543 2024-03-06T10:59:52Z ddc:004 open_access

Exact Completeness of LP Hierarchies for Linear Codes Coregliano, Leonardo Nagami Jeronimo, Fernando Granha Jones, Chris LP bound linear codes Delsarte’s LP combinatorial polytopes pseudoexpectation Determining the maximum size A₂(n,d) of a binary code of blocklength n and distance d remains an elusive open question even when restricted to the important class of linear codes. Recently, two linear programming hierarchies extending Delsarte’s LP were independently proposed to upper bound A₂^{Lin}(n,d) (the analogue of A₂(n,d) for linear codes). One of these hierarchies, by the authors, was shown to be approximately complete in the sense that the hierarchy converges to A₂^{Lin}(n,d) as the level grows beyond n². Despite some structural similarities, not even approximate completeness was known for the other hierarchy by Loyfer and Linial. In this work, we prove that both hierarchies recover the exact value of A₂^{Lin}(n,d) at level n. We also prove that at this level the polytope of Loyfer and Linial is integral. Even though these hierarchies seem less powerful than general hierarchies such as Sum-of-Squares, we show that they have enough structure to yield exact completeness via pseudoprobabilities. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Leonardo Nagami Coregliano and Fernando Granha Jeronimo and Chris Jones 2023 Is Part Of LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ITCS.2023.40 urn:nbn:de:0030-drops-175433 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.40 eng https://creativecommons.org/licenses/by/4.0/legalcode