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High-entropy dual functions over finite fields and locally decodable codes

Part of: Sequences and sets

Published online by Cambridge University Press:  08 March 2021

Jop Briët  and
Farrokh Labib
Jop Briët
Affiliation:
CWI, Science Park 123, 1098 XGAmsterdam, The Netherlands; E-mails: j.briet@cwi.nl and labib@cwi.nl.
Farrokh Labib
Affiliation:
CWI, Science Park 123, 1098 XGAmsterdam, The Netherlands; E-mails: j.briet@cwi.nl and labib@cwi.nl.

Abstract

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We show that for infinitely many primes p there exist dual functions of order k over ${\mathbb{F}}_p^n$ that cannot be approximated in $L_\infty $-distance by polynomial phase functions of degree $k-1$. This answers in the negative a natural finite-field analogue of a problem of Frantzikinakis on $L_\infty $-approximations of dual functions over ${\mathbb{N}}$ (a.k.a. multiple correlation sequences) by nilsequences.

MSC classification

Primary: 11B30: Arithmetic combinatorics; higher degree uniformity
Secondary: 11B25: Arithmetic progressions
Type
Analysis
Information
Forum of Mathematics, Sigma , Volume 9 , 2021 , e19
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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