Outlaw distributions and locally decodable codes

Locally decodable codes (LDCs) are error correcting codes that allow for decoding of a single message bit using a small number of queries to a corrupted encoding. Despite decades of study, the optimal trade-off between query complexity and codeword length is far from understood. In this work, we give 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. We coin the term `outlaw distributions' for such distributions since they `defy' the Law of Large Numbers. We show that the existence of outlaw distributions over sufficiently `smooth' functions implies the existence of constant query LDCs and vice versa. We give several candidates for outlaw distributions over smooth functions coming from finite field incidence geometry, additive combinatorics and from hypergraph (non)expanders. We also prove a useful lemma showing that (smooth) LDCs which are only required to work on average over a random message and a random message index can be turned into true LDCs at the cost of only constant factors in the parameters.Comment: A preliminary version of this paper appeared in the proceedings of ITCS 201

Outlaw distributions and locally decodable codes

Locally decodable codes (LDCs) are error correcting codes that allow for decoding of a single message bit using a small number of queries to a corrupted encoding. Despite decades of study, the optimal trade-off between query complexity and codeword length is far from understood. In this work, we give a new characterization of LDCs using distributions over Boolean functions whose expectation is hard to approximate (in L∞ norm) with a small number of samples. We coin the term “outlaw distributions” for such distributions since they “defy” the Law of Large Numbers. We show that the existence of outlaw distributions over sufficiently “smooth” functions implies the existence of constant query LDCs and vice versa. We give several candidates for outlaw distributions over smooth functions coming from finite field incidence geometry, additive combinatorics and hypergraph (non)expanders. We also prove a useful lemma showing that (smooth) LDCs which are only required to work on average over a random message and a random message index can be turned into true LDCs at the cost of only constant factors in the parameters

Gaussian width bounds with applications to arithmetic progressions in random settings

Motivated by problems on random differences in Szemer\'{e}di's theorem and on large deviations for arithmetic progressions in random sets, we prove upper bounds on the Gaussian width of point sets that are formed by 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. We show the following applications of our bounds. Let $[\mathbb{Z}/N\mathbb{Z}]_p$ be the random subset of $\mathbb{Z}/N\mathbb{Z}$ containing each element independently with probability $p$. $\bullet$ A set $D\subseteq \mathbb{Z}/N\mathbb{Z}$ is $\ell$-intersective if any dense subset of $\mathbb{Z}/N\mathbb{Z}$ contains a proper $(\ell+1)$-term arithmetic progression with common difference in $D$. Our main result implies that $[\mathbb{Z}/N\mathbb{Z}]_p$ is $\ell$-intersective with probability $1 - o(1)$ provided $p \geq \omega(N^{-\beta_\ell}\log N)$ for $\beta_\ell = (\lceil(\ell+1)/2\rceil)^{-1}$. This gives a polynomial improvement for all $\ell \ge 3$ of a previous bound due to Frantzikinakis, Lesigne and Wierdl, and reproves more directly the same improvement shown recently by the authors and Dvir. $\bullet$ Let $X_k$ be the number of $k$-term arithmetic progressions in $[\mathbb{Z}/N\mathbb{Z}]_p$ and consider the large deviation rate $\rho_k(\delta) = \log\Pr[X_k \geq (1+\delta)\mathbb{E}X_k]$. We give quadratic improvements of the best-known range of $p$ for which a highly precise estimate of $\rho_k(\delta)$ due to Bhattacharya, Ganguly, Shao and Zhao is valid for all odd $k \geq 5$. We also discuss connections with error correcting codes (locally decodable codes) and the Banach-space notion of type for injective tensor products of $\ell_p$-spaces.Comment: 18 pages, some typos fixe

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

We show that for infinitely many primes p, there exist dual functions of order k over Fnp that cannot be approximated in L∞-distance by polynomial phase functions of degree k−1. This answers in the negative a natural finite-field analog of a problem of Frantzikinakis on L∞-approximations of dual functions over N (a.k.a. multiple correlation sequences) by nilsequences

Gaussian width bounds with applications to arithmetic progressions in random settings

Motivated by two problems on arithmetic progressions (APs)—concerning large deviations for AP counts in random sets and random differences in Szemer´edi’s theorem— we prove upper bounds on the Gaussian width of the image of the n-dimensional Boolean hypercube under a mapping ψ : Rn → Rk, where each coordinate is a constant-degree multilinear polynomial with 0/1 coefficients. We show the following applications of our bounds. Let [Z/NZ]p be the random subset of Z/NZ containing each element independently with probability p. • Let Xk be the number of k-term APs in [Z/NZ]p. We show that a precise estimate on the large deviation rate log Pr[Xk ≥ (1 + δ)EXk] due to Bhattacharya, Ganguly, Shao and Zhao is valid if

A Lower Bound for Relaxed Locally Decodable Codes

A locally decodable code (LDC) C \colon \bitset^k \to \bitset^n is an error correcting code wherein individual bits of the message can be recovered by only querying a few bits of a noisy codeword. LDCs found a myriad of applications both in theory and in practice, ranging from probabilistically checkable proofs to distributed storage. However, despite nearly two decades of extensive study, the best known constructions of $O(1)$-query LDCs have super-polynomial blocklength. The notion of relaxed LDCs is a natural relaxation of LDCs, which aims to bypass the foregoing barrier by requiring local decoding of nearly all individual message bits, yet allowing decoding failure (but not error) on the rest. State of the art constructions of $O(1)$-query relaxed LDCs achieve blocklength $n = O\left(k^{1+ \gamma}\right)$ for an arbitrarily small constant $\gamma$. We prove a lower bound which shows that $O(1)$-query relaxed LDCs cannot achieve blocklength $n = k^{1+ o(1)}$. This resolves an open problem raised by Goldreich in 2004

Raising the roof on the threshold for Szemerédi‘s theorem with random differences

Using recent developments on the theory of locally decodable codes, we prove that the critical size for Szemerédi’s theorem with random differences is bounded from above by N 1 − 2 k + o (1) for length- k progressions. This improves the previous best bounds of N 1 − 1 d k/ 2 e + o (1) for all odd k

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

We show that for infinitely many primes p, there exist dual functions of order k over Fnp that cannot be approximated in L∞-distance by polynomial phase functions of degree k−1. This answers in the negative a natural finite-field analog of a problem of Frantzikinakis on L∞-approximations of dual functions over N (a.k.a. multiple correlation sequences) by nilsequences

On the power of relaxed Local Decoding Algorithms

A locally decodable code (LDC) C from {0,1} to the k to {0,1} to the n is an error correcting code wherein individual bits of the message can be recovered by only querying a few bits of a noisy codeword. LDCs found a myriad of applications both in theory and in practice, ranging from probabilistically checkable proofs to distributed storage. However, despite nearly two decades of extensive study, the best known constructions of O(1)-query LDCs have super-polynomial block length. The notion of relaxed LDCs is a natural relaxation of LDCs, which aims to bypass the foregoing barrier by requiring local decoding of nearly all individual message bits, yet allowing decoding failure (but not error) on the rest. State of the art constructions of O(1)-query relaxed LDCs achieve blocklength n is order of k to the power of 1 plus gamma for an arbitrarily small constant. We prove a lower bound which shows that O(1)-query relaxed LDCs cannot achieve blocklength n = k to the power of 1 + o(1). This resolves an open problem raised by Goldreich in 2004