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TR19-080 | 1st June 2019 17:00

#### On List Recovery of High-Rate Tensor Codes

TR19-080
Authors: Swastik Kopparty, Nicolas Resch, Noga Ron-Zewi, Shubhangi Saraf, Shashwat Silas
Publication: 1st June 2019 19:43
Keywords:

Abstract:

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 {\em approximately} locally list recoverable, as well as globally list recoverable in {\em 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) {\em probabilistic} near-linear time global list decoding algorithms. This was also yielded constant-rate codes approaching the Gilbert-Varshamov bound with {\em 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 {\em deterministic} near-linear time. This yields in turn the first capacity-achieving list decodable codes with {\em deterministic} near-linear time global list decoding algorithms. It also gives constant-rate codes approaching the Gilbert Varshamov bound with {\em 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 Gilbert-Varshamov bound that are {\em locally correctable} with query complexity and running time $N^{o(1)}$. This improves over prior work by Gopi et. al. (SODA'17; IEEE Transactions on Information Theory'18) that only gave query complexity $N^{\epsilon}$ with rate that is exponentially small in $1/\epsilon$.

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 $N^{\Omega(1/\log \log N)}$ on the product of query complexity and output list size for locally list recovering high-rate tensor codes.

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