Relaxed Locally Correctable Codes with Improved Parameters

Abstract

Locally decodable codes (LDCs) are error-correcting codes C : Σ k → Σ n that admit a local decoding algorithm that recovers each individual bit of the message by querying only a few bits from a noisy codeword. An important question in this line of research is to understand the optimal trade-off between the query complexity of LDCs and their block length. Despite importance of these objects, the best known constructions of constant query LDCs have super-polynomial length, and there is a significant gap between the best constructions and the known lower bounds in terms of the block length.For many applications it suffices to consider the weaker notion of relaxed LDCs (RLDCs), which allows the local decoding algorithm to abort if by querying a few bits it detects that the input is not a codeword. This relaxation turned out to allow decoding algorithms with constant query complexity for codes with almost linear length. Specifically, [BGH + 06] constructed an O(q)-query RLDC that encodes a message of length k using a codeword of block length n = O(k 1+1/ √ q ).In this work we improve the parameters of [BGH + 06] by constructing an O(q)-query RLDC that encodes a message of length k using a codeword of block length O(k 1+1/q ). This construction matches (up to a multiplicative constant factor) the lower bounds of [KT00, Woo07] for constant query LDCs, thus making progress toward understanding the gap between LDCs and RLDCs in the constant query regime.In fact, our construction extends to the stronger notion of relaxed locally correctable codes (RLCCs), introduced in [GRR18], where given a noisy codeword the correcting algorithm either recovers each individual bit of the codeword by only reading a small part of the input, or aborts if the input is detected to be corrupt.

A Structural Theorem for Local Algorithms with Applications to Coding, Testing, and Privacy

“…This shows that O(1)-query relaxed LDCs cannot ob- 3 As observed in [5], these two conditions suffice for obtaining a third condition which guarantees that the decoder only outputs ⊥ on an arbitrarily small fraction of the coordinates.…”

“…More accurately, a relaxed LDC C : {0, 1} k → {0, 1} n with decoding radius δ is a code that admits a probabilistic algorithm, a decoder, which on index i ∈ [k] makes queries to a string w ∈ {0, 1} n that is δ-close to a codeword C(x) and satisfies the following: (1) if the input is a valid codeword (i.e., w = C(x)), the decoder outputs x i with high probability; and (2) otherwise, with high probability, the decoder must either output x i or a special "abort" symbol ⊥, indicating it detected an error and is unable to decode. 3 This seemingly modest relaxation allows for obtaining dramatically stronger parameters. Indeed, Ben-Sasson et al [5] constructed a q-query relaxed LDC with blocklength n = k 1+1/Ω( √ q) , and raised the problem of whether it is possible to obtain a better rate; the best known construction, obtained in recent work of Asadi and Shinkar [3], improves it to n = k 1+1/Ω(q) .…”

“…3 This seemingly modest relaxation allows for obtaining dramatically stronger parameters. Indeed, Ben-Sasson et al [5] constructed a q-query relaxed LDC with blocklength n = k 1+1/Ω( √ q) , and raised the problem of whether it is possible to obtain a better rate; the best known construction, obtained in recent work of Asadi and Shinkar [3], improves it to n = k 1+1/Ω(q) . We stress that proving lower bounds on relaxed LDCs is significantly harder than on standard LDCs, and indeed, the first non-trivial lower bound was only recently obtained in [34], which shows that, to obtain query complexity q, the code must have blocklength…”