A Complete Linear Programming Hierarchy 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 would like to thank the anonymous reviewers for their comments and helpful feedback.

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Leonardo Nagami Coregliano, Fernando Granha Jeronimo, and Chris Jones. A Complete Linear Programming Hierarchy for Linear Codes. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 51:1-51:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


A longstanding open problem in coding theory is to determine the best (asymptotic) rate R₂(δ) of binary codes with minimum constant (relative) distance δ. An existential lower bound was given by Gilbert and Varshamov in the 1950s. On the impossibility side, in the 1970s McEliece, Rodemich, Rumsey and Welch (MRRW) proved an upper bound by analyzing Delsarte’s linear programs. To date these results remain the best known lower and upper bounds on R₂(δ) with no improvement even for the important class of linear codes. Asymptotically, these bounds differ by an exponential factor in the blocklength. In this work, we introduce a new hierarchy of linear programs (LPs) that converges to the true size A^{Lin}₂(n,d) of an optimum linear binary code (in fact, over any finite field) of a given blocklength n and distance d. This hierarchy has several notable features: 1) It is a natural generalization of the Delsarte LPs used in the first MRRW bound. 2) It is a hierarchy of linear programs rather than semi-definite programs potentially making it more amenable to theoretical analysis. 3) It is complete in the sense that the optimum code size can be retrieved from level O(n²). 4) It provides an answer in the form of a hierarchy (in larger dimensional spaces) to the question of how to cut Delsarte’s LP polytopes to approximate the true size of linear codes. We obtain our hierarchy by generalizing the Krawtchouk polynomials and MacWilliams inequalities to a suitable "higher-order" version taking into account interactions of 𝓁 words. Our method also generalizes to translation schemes under mild assumptions.

Subject Classification

ACM Subject Classification
  • Theory of computation
  • Coding theory
  • code bounds
  • convex programming
  • linear programming hierarchy


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