Outlaw distributions and locally decodable codes

TL;DR

This paper introduces outlaw distributions as a new way to understand the limits of locally decodable codes, connecting their existence to properties of certain Boolean function distributions and providing new constructions and transformations.

Contribution

It provides a novel characterization of LDCs using outlaw distributions, links their existence to smooth functions from geometry and combinatorics, and offers a method to convert average-case LDCs into full LDCs.

Findings

Existence of outlaw distributions implies constant query LDCs.

Several candidate outlaw distributions are identified from geometric and combinatorial sources.

Average-case LDCs can be transformed into true LDCs with only constant parameter loss.

Abstract

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.