Forbidden intersections for codes

Author:

Keevash Peter1,Lifshitz Noam2,Long Eoin3,Minzer Dor4ORCID

Affiliation:

1. Mathematical Institute University of Oxford UK

2. Einstein Institute of Mathematics Hebrew University Jerusalem Israel

3. School of Mathematics University of Birmingham Birmingham UK

4. Department of Mathematics Massachusetts Institute of Technology Cambridge USA

Abstract

AbstractDetermining the maximum size of a ‐intersecting code in was a longstanding open problem of Frankl and Füredi, solved independently by Ahlswede and Khachatrian and by Frankl and Tokushige. We extend their result to the setting of forbidden intersections, by showing that for any and large compared with (but not necessarily ) that the same bound holds for codes with the weaker property of being ‐avoiding, that is, having no two vectors that agree on exactly coordinates. Our proof proceeds via a junta approximation result of independent interest, which we prove via a development of our recent theory of global hypercontractivity: we show that any ‐avoiding code is approximately contained in a ‐intersecting junta (a code where membership is determined by a constant number of coordinates). In particular, when , this gives an alternative proof of a recent result of Eberhard, Kahn, Narayanan and Spirkl that symmetric intersecting codes in have size .

Funder

European Research Council

Publisher

Wiley

Subject

General Mathematics

Cited by 2 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Stability for the complete intersection theorem, and the forbidden intersection problem of Erdős and Sós;Journal of the European Mathematical Society;2024-02-21

2. Optimal Testing of Generalized Reed-Muller Codes in Fewer Queries;2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS);2023-11-06

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