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.
Highlights
Error correcting codes (ECCs) solve the basic problem of communication over noisy channels by encoding a message into a codeword from which, even if the channel partially corrupts it, the message can later be retrieved
We prove a useful lemma showing that Locally decodable codes (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
The defining feature of LDCs is that they allow for ultrafast decoding of single message bits, a property that typical ECCs lack, as their decoders must read an entire codeword to achieve the same
Summary
Error correcting codes (ECCs) solve the basic problem of communication over noisy channels by encoding a message into a codeword from which, even if the channel partially corrupts it, the message can later be retrieved. The defining feature of LDCs is that they allow for ultrafast decoding of single message bits, a property that typical ECCs lack, as their decoders must read an entire (possibly corrupted) codeword to achieve the same They were first formally defined in the context of channel coding in [25], they (and the closely related locally correctable codes) implicitly appeared in several previous works in other settings, such as computation and program checking [4, 5], probabilistically checkable proofs [3, 2] and private information retrieval schemes (PIRs) [11]. We cannot yet rule out the exciting possibility that constant rate LDCs with polylogarithmic query complexity exist
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