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Higher-Order Delsarte Dual LPs: Lifting, Constructions and Completeness

Authors Leonardo Nagami Coregliano , Fernando Granha Jeronimo , Chris Jones , Nati Linial , Elyassaf Loyfer

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ACM Subject Classification
  • Theory of computation → Error-correcting codes
Keywords
  • Coding theory
  • code bounds
  • convex optimization
  • linear progamming hierarchy

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Abstract

A central and longstanding open problem in coding theory is the rate-versus-distance trade-off for binary error-correcting codes. In a seminal work, Delsarte introduced a family of linear programs establishing relaxations on the size of optimum codes. To date, the state-of-the-art upper bounds for binary codes come from dual feasible solutions to these LPs. Still, these bounds are exponentially far from the best-known existential constructions.
Recently, hierarchies of linear programs extending and strengthening Delsarte’s original LPs were introduced for linear codes, which we refer to as higher-order Delsarte LPs. These new hierarchies were shown to provably converge to the actual value of optimum codes, namely, they are complete hierarchies. Therefore, understanding them and their dual formulations becomes a valuable line of investigation. Nonetheless, their higher-order structure poses challenges. In fact, analysis of all known convex programming hierarchies strengthening Delsarte’s original LPs has turned out to be exceedingly difficult and essentially nothing is known, stalling progress in the area since the 1970s.
Our main result is an analysis of the higher-order Delsarte LPs via their dual formulation. Although quantitatively, our current analysis only matches the best-known upper bounds, it shows, for the first time, how to tame the complexity of analyzing a hierarchy strengthening Delsarte’s original LPs. In doing so, we reach a better understanding of the structure of the hierarchy, which may serve as the foundation for further quantitative improvements. We provide two additional structural results for this hierarchy. First, we show how to explicitly lift any feasible dual solution from level k to a (suitable) larger level 𝓁 while retaining the objective value. Second, we give a novel proof of completeness using the dual formulation.

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Leonardo Nagami Coregliano, Fernando Granha Jeronimo, Chris Jones, Nati Linial, and Elyassaf Loyfer. Higher-Order Delsarte Dual LPs: Lifting, Constructions and Completeness. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 44:1-44:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ITCS.2026.44

Author Details

Leonardo Nagami Coregliano
  • University of Chicago, IL, USA
Fernando Granha Jeronimo
  • University of Illinois Urbana-Champaign, IL, USA
Chris Jones
  • University of California, Davis, CA, USA
Nati Linial
  • The Hebrew University of Jerusalem, Israel
Elyassaf Loyfer
  • The Hebrew University of Jerusalem, Israel

Funding

Work supported in part by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement Nos. 834861 and 101019547).
  • Linial, Nati: Supported in part by an ERC Grant 101141253, "Packing in Discrete Domains - Geometry and Analysis".

Acknowledgements

The authors are very thankful for the support and hospitality of IAS and the Simons Institute. LC, FGJ and EL thank Avi Wigderson for hosting them at the wonderful IAS, where part of this work was done. In particular, we would like to highlight the importance to us of the programs and clusters: "HDX and Codes", "Analysis and TCS: New Frontiers", "Error-Correcting Codes: Theory and Practice", and "Quantum Algorithms, Complexity, and Fault Tolerance". FGJ thanks Venkat Guruswami for kindly hosting him in his fantastic research groups. FGJ thanks the support in part as a Google Research Fellow. CJ is a member of the Bocconi Institute for Data Science and Analytics (BIDSA).

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