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

@article{Brit2021HighentropyDF,
  title={High-entropy dual functions over finite fields and locally decodable codes},
  author={Jop Bri{\"e}t and Farrokh Labib},
  journal={Forum of Mathematics, Sigma},
  year={2021},
  volume={9}
}
Abstract 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. 
2 Citations
High-Entropy Dual Functions and Locally Decodable Codes (Extended Abstract)
TLDR
Szemerédi’s theorem with random differences, in particular lower bounds on ρk, can be used to show the existence of LDCs, and the finite-field conjecture is motivated mainly by the possibility that so-called dual functions can be approximated well by polynomial phases. Expand
Multiple correlation sequences not approximable by nilsequences
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References

SHOWING 1-10 OF 53 REFERENCES
The Inverse Conjecture for the Gowers Norm over Finite Fields in Low Characteristic
We establish the inverse conjecture for the Gowers norm over finite fields, which asserts (roughly speaking) that if a bounded function $${f : V \rightarrow \mathbb{C}}$$ on a finite-dimensionalExpand
Good cyclic codes and the uncertainty principle
TLDR
It is pointed out that even a weak version of the uncertainty principle for fields of positive characteristic would imply that good cyclic codes do exist and some heuristic arguments supporting that this is indeed the case are provided. Expand
Random Sequences and Pointwise Convergence of Multiple Ergodic Averages
We prove pointwise convergence, as $N\to \infty$, for the multiple ergodic averages $\frac{1}{N}\sum_{n=1}^N f(T^nx)\cdot g(S^{a_n}x)$, where $T$ and $S$ are commuting measure preservingExpand
Locally decodable codes and the failure of cotype for projective tensor products
It is shown that for every $p\in (1,\infty)$ there exists a Banach space $X$ of finite cotype such that the projective tensor product $l_p\hat\otimes X$ fails to have finite cotype. More generally,Expand
Gaussian width bounds with applications to arithmetic progressions in random settings
  • J. Briët, S. Gopi
  • Mathematics, Computer Science
  • International Mathematics Research Notices
  • 2018
TLDR
Upper bounds are proved on the Gaussian width of the image of the $n$-dimensional Boolean hypercube under a mapping $\psi:\mathbb{R}^n\to\mathbb(R)^k$, where each coordinate is a constant-degree multilinear polynomial with $0/1$ coefficients. Expand
On Szemerédi’s theorem with differences from a random set
We consider, over both the integers and finite fields, Szemeredi's theorem on $k$-term arithmetic progressions where the set $S$ of allowed common differences in those progressions is restricted andExpand
Outlaw Distributions and Locally Decodable Codes
TLDR
This work gives a new characterization of LDCs using distributions over Boolean functions whose expectation is hard to approximate (in~$L_\infty$~norm) with a small number of samples and coin the term `outlaw distributions' for such distributions since they `defy' the Law of Large Numbers. Expand
Multiple correlation sequences not approximable by nilsequences
<jats:p>We show that there is a measure-preserving system <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png"Expand
The inverse conjecture for the Gowers norm over finite fields via the correspondence principle
The inverse conjecture for the Gowers norms U d .V/ for finite-dimensional vector spaces V over a finite field F asserts, roughly speaking, that a bounded function f has large Gowers normk fkU d.V/Expand
On a problem of Arnold: The average multiplicative order of a given integer
For coprime integers g and n, let l(g) (n) denote the multiplicative order of g modulo n. Motivated by a conjecture of Arnold, we study the average of l(g) (n) as n <= x ranges over integers coprimeExpand
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