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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 ∞ 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.

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

“…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 [BDG19], 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 alluded to above.…”

“…In [BDG19] it is shown that there exist of 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.…”