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 approximately locally list recoverable, as well as globally list recoverable in probabilistic 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) probabilistic near-linear time global list decoding algorithms. This was also yielded constant-rate codes approaching the Gilbert-Varshamov bound with probabilistic 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 deterministic near-linear time. This yields in turn the first capacity-achieving list decodable codes with deterministic near-linear time global list decoding algorithms. It also gives constant-rate codes approaching the Gilbert-Varshamov bound with deterministic 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 constant-rate codes approaching the GilbertVarshamov bound that are locally correctable with query complexity and running time N. This improves over prior work by Gopi et. al. (SODA’17; IEEE Transactions on Information Theory’18) that only gave query complexity N with rate that is exponentially small in 1/ε. 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 N log log N) on the product of query complexity and output list size for locally list recovering high-rate tensor codes. ∗Department of Mathematics and Department of Computer Science, Rutgers University. swastik.kopparty@gmail.com. †Department of Computer Science, Carnegie Mellon University. nresch@andrew.cmu.edu. ‡Department of Computer Science, University of Haifa. noga@cs.haifa.ac.il. §Department of Mathematics and Department of Computer Science, Rutgers University. shubhangi.saraf@gmail.com. ¶Department of Computer Science, Stanford University. silas@stanford.edu .

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