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# Candidate Tree Codes via Pascal Determinant Cubes

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## Acknowledgements

The second author wishes to thank Roni Con, Shir Peleg-Schatzman, Noam Peri, Tal Roth, and Shahar Samocha for interesting discussions on tree codes.

## Cite As

Inbar Ben Yaacov, Gil Cohen, and Anand Kumar Narayanan. Candidate Tree Codes via Pascal Determinant Cubes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 54:1-54:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2021.54

## Abstract

Tree codes are combinatorial structures introduced by Schulman [Schulman, 1993] 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 [Moore and Schulman, 2014]. 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 [Cohen et al., 2018], 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. [Gelles et al., 2016] 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.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Error-correcting codes
##### Keywords
• Tree codes
• Sparse polynomials
• Explicit constructions

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