Candidate tree codes via pascal determinant cubes

Inbar Ben Yaacov, Gil Cohen, Anand Kumar Narayanan

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

Tree codes are combinatorial structures introduced by Schulman [23] as key ingredients in interactive coding schemes. Asymptotically-good tree codes are long known to exist, yet their explicit construction remains a notoriously hard open problem. Even proposing a plausible construction, without the burden of proof, is difficult and the defining tree code property requires structure that remains elusive. To the best of our knowledge, only one candidate appears in the literature, due to Moore and Schulman [19]. We put forth a new candidate for an explicit asymptotically-good tree code. Our construction is an extension of the vanishing rate tree code by Cohen-Haeupler-Schulman [7], and its correctness relies on a conjecture that we introduce on certain Pascal determinants indexed by the points of the Boolean hypercube. Furthermore, using the vanishing distance tree code by Gelles et al. [12] enables us to present a construction that relies on an even weaker assumption. We furnish evidence supporting our conjecture through numerical computation, combinatorial arguments from planar path graphs and based on well-studied heuristics from arithmetic geometry.

Original languageEnglish
Title of host publicationApproximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2021
EditorsMary Wootters, Laura Sanita
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772075
DOIs
StatePublished - 1 Sep 2021
Externally publishedYes
Event24th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2021 and 25th International Conference on Randomization and Computation, RANDOM 2021 - Virtual, Seattle, United States
Duration: 16 Aug 202118 Aug 2021

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume207
ISSN (Print)1868-8969

Conference

Conference24th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2021 and 25th International Conference on Randomization and Computation, RANDOM 2021
Country/TerritoryUnited States
CityVirtual, Seattle
Period16/08/2118/08/21

Keywords

  • Explicit constructions
  • Sparse polynomials
  • Tree codes

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