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Erasure codes and Turán hypercube problems

Noga Alon

TL;DR

It is observed that several vertex Tur\'an type problems for the hypercube are equivalent to questions about erasure list-decodable codes and improved bounds are obtained for some of these problems.

Abstract

We observe that several vertex Turán type problems for the hypercube that received a considerable amount of attention in the combinatorial community are equivalent to questions about erasure list-decodable codes. Analyzing a recent construction of Ellis, Ivan and Leader, and determining the Turán density of certain hypergraph augemntations we obtain improved bounds for some of these problems.

Erasure codes and Turán hypercube problems

TL;DR

It is observed that several vertex Tur\'an type problems for the hypercube are equivalent to questions about erasure list-decodable codes and improved bounds are obtained for some of these problems.

Abstract

We observe that several vertex Turán type problems for the hypercube that received a considerable amount of attention in the combinatorial community are equivalent to questions about erasure list-decodable codes. Analyzing a recent construction of Ellis, Ivan and Leader, and determining the Turán density of certain hypergraph augemntations we obtain improved bounds for some of these problems.
Paper Structure (11 sections, 11 theorems, 18 equations)

This paper contains 11 sections, 11 theorems, 18 equations.

Table of Contents

  1. Introduction and results
  2. Erasure codes and Turán hypercube problems
  3. $(d,2^d-1,n)$-codes
  4. Codes of positive density
  5. Graph and hypergraph augmentations
  6. Proofs
  7. Augmentations
  8. Hitting subcubes
  9. List Erasure Codes
  10. Lagrangians
  11. Concluding remarks

Key Result

Proposition 1.1

For every large $k$ and every $n$ there is a subset of less than a fraction of $2^{-k}$ of the vertices of the $n$-cube that intersects the set of vertices of any cube of dimension $d=2k + 3 \log_2 k$.

Theorems & Definitions (18)

  • Proposition 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Proposition 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 8 more