Exact Completeness of LP Hierarchies for Linear Codes

Authors Leonardo Nagami Coregliano , Fernando Granha Jeronimo, Chris Jones

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Author Details

Leonardo Nagami Coregliano
  • Institute for Advanced Study, Princeton, NJ, USA
Fernando Granha Jeronimo
  • Institute for Advanced Study, Princeton, NJ, USA
Chris Jones
  • University of Chicago, Il, USA


We thank Avi Wigderson for stimulating discussions during the initial phase of this project.

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Leonardo Nagami Coregliano, Fernando Granha Jeronimo, and Chris Jones. Exact Completeness of LP Hierarchies for Linear Codes. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 40:1-40:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Coding theory
  • Mathematics of computing → Combinatorial optimization
  • LP bound
  • linear codes
  • Delsarte’s LP
  • combinatorial polytopes
  • pseudoexpectation


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