On List Recovery of High-Rate Tensor Codes

We continue the study of list recovery properties of high-rate tensor codes, initiated by Hemenway, Ron-Zewi, and Wootters (FOCS’17). In that work it was shown that the tensor product of an efficient (poly-time) high-rate globally list recoverable code is <italic>approximately</italic> locally list recoverable, as well as globally list recoverable in <italic>probabilistic</italic> near-linear time. This was used in turn to give the first capacity-achieving list decodable codes with (1) local list decoding algorithms, and with (2) <italic>probabilistic</italic> near-linear time global list decoding algorithms. This also yielded constant-rate codes approaching the Gilbert-Varshamov bound with <italic>probabilistic</italic> near-linear time global unique decoding algorithms. In the current work we obtain the following results: 1) The tensor product of an efficient (poly-time) high-rate globally list recoverable code is globally list recoverable in <italic>deterministic</italic> near-linear time. This yields in turn the first capacity-achieving list decodable codes with <italic>deterministic</italic> near-linear time global list decoding algorithms. It also gives constant-rate codes approaching the Gilbert-Varshamov bound with <italic>deterministic</italic> near-linear time global unique decoding algorithms. 2) If the base code is additionally locally correctable, then the tensor product is (genuinely) locally list recoverable. This yields in turn (non-explicit) constant-rate codes approaching the Gilbert-Varshamov bound that are <italic>locally correctable</italic> with query complexity and running time <inline-formula> <tex-math notation="LaTeX">$N^{o(1)}$ </tex-math></inline-formula>. This improves over prior work by Gopi et. al. (SODA’17; IEEE Transactions on Information Theory’18) that only gave query complexity <inline-formula> <tex-math notation="LaTeX">$N^{ \varepsilon }$ </tex-math></inline-formula> with rate that is exponentially small in <inline-formula> <tex-math notation="LaTeX">$1/ \varepsilon $ </tex-math></inline-formula>. 3) A nearly-tight combinatorial lower bound on output list size for list recovering high-rate tensor codes. This bound implies in turn a nearly-tight lower bound of <inline-formula> <tex-math notation="LaTeX">$N^{\Omega (1/\log \log N)}$ </tex-math></inline-formula> on the product of query complexity and output list size for locally list recovering high-rate tensor codes.

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