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DOI: 10.4230/LIPIcs.ITCS.2021.76
URN: urn:nbn:de:0030-drops-136157
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Briët, Jop ; Labib, Farrokh

High-Entropy Dual Functions and Locally Decodable Codes (Extended Abstract)

LIPIcs-ITCS-2021-76.pdf (0.2 MB)


Locally decodable codes (LDCs) allow any single encoded message symbol to be retrieved from a codeword with good probability by reading only a tiny number of codeword symbols, even if the codeword is partially corrupted. LDCs have surprisingly many applications in computer science and mathematics (we refer to [Yekhanin, 2012; Lovett, 2007] for extensive surveys). But despite their ubiquity, they are poorly understood. Of particular interest is the tradeoff between the codeword length N as a function of message length k when the query complexity - the number of probed codeword symbols - and alphabet size are constant. The Hadamard code is a 2-query LDC of length N = 2^O(k) and this length is optimal in the 2-query regime [Lovett, 2007]. For q ≥ 3, near-exponential gaps persist between the best-known upper and lower bounds. The family of Reed-Muller codes, which generalize the Hadamard code, were for a long time the best-known examples, giving q-query LDCs of length exp(O(k^{1/(q-1)})), until breakthrough constructions of matching vector LDCs of Yekhanin and Efremenko [Yekhanin, 2008; Efremenko, 2012].
In contrast with other combinatorial objects such as expander graphs, the probabilistic method has so far not been successfully used to beat the best explicit LDC constructions. In [Lovett, 2007], a probabilistic framework was given that could in principle yield best-possible LDCs, albeit non-constructively. A special instance of this framework connects LDCs with a probabilistic version of Szemerédi’s theorem. The setup for this is as follows: For a finite abelian group G of size N = |G|, let D ⊆ G be a random subset where each element is present with probability ρ independently of all others. For k ≥ 3 and ε ∈ (0,1), let E be the event that every subset A ⊆ G of size |A| ≥ ε |G| contains a proper k-term arithmetic progression with common difference in D. For fixed ε > 0 and sufficiently large N, it is an open problem to determine the smallest value of ρ - denoted ρ_k - such that Pr[E] ≥ 1/2. In [Lovett, 2007] it is shown that there exist k-query LDCs of message length Ω(ρ_k N) and codeword length O(N). As such, Szemerédi’s theorem with random differences, in particular lower bounds on ρ_k, can be used to show the existence of LDCs. Conversely, this connection indirectly implies the best-known upper bounds on ρ_k for all k ≥ 3 [Lovett, 2007; Lovett, 2007]. However, a conjecture from [Lovett, 2007] states that over ℤ_N we have ρ_k ≤ O_k(N^{-1}log N) for all k, which would be best-possible. Truth of this conjecture would imply that over this group, Szemerédi’s theorem with random differences cannot give LDCs better than the Hadamard code. For finite fields, Altman [Lovett, 2007] showed that this is false. In particular, over 𝔽_pⁿ for p odd, he proved that ρ₃ ≥ Ω(p^{-n} n²); generally, ρ_k ≥ Ω(p^{-n} n^{k-1}) holds when p ≥ k+1 [Lovett, 2007]. In turn, these bounds are conjectured to be optimal for the finite-field setting, which would imply that over finite fields, Szemerédi’s theorem with random differences cannot give LDCs better than Reed-Muller codes.
The finite-field conjecture is motivated mainly by the possibility that so-called dual functions can be approximated well by polynomial phases, functions of the form e^{2π i P(x)/p} where P is a multivariate polynomial over 𝔽_p. We show that this is false. Using Yekhanin’s matching-vector-code construction, we give dual functions of order k over 𝔽_pⁿ that cannot be approximated in L_∞-distance by polynomial phases of degree k-1. This answers in the negative a natural finite-field analog of a problem of Frantzikinakis over ℕ [Lovett, 2007].

BibTeX - Entry

  author =	{Jop Bri\"{e}t and Farrokh Labib},
  title =	{{High-Entropy Dual Functions and Locally Decodable Codes}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{76:1--76:2},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{James R. Lee},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-136157},
  doi =		{10.4230/LIPIcs.ITCS.2021.76},
  annote =	{Keywords: Higher-order Fourier analysis, dual functions, finite fields, additive combinatorics, coding theory}

Keywords: Higher-order Fourier analysis, dual functions, finite fields, additive combinatorics, coding theory
Collection: 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)
Issue Date: 2021
Date of publication: 04.02.2021

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