Local List Recovery of High-Rate Tensor Codes & Applications

Abstract

In this work, we give the first construction of high-rate locally list-recoverable codes. Listrecovery has been an extremely useful building block in coding theory, and our motivation is to use these codes as such a building block. In particular, our construction gives the first capacity-achieving locally list-decodable codes (over constant-sized alphabet); the first capacity achieving globally list-decodable codes with nearly linear time list decoding algorithm (once more, over constant-sized alphabet); and a randomized construction of binary codes on the Gilbert-Varshamov bound that can be uniquely decoded in near-linear-time, with higher rate than was previously known.Our techniques are actually quite simple, and are inspired by an approach of Gopalan, Guruswami, and Raghavendra (Siam Journal on Computing, 2011) for list-decoding tensor codes. We show that tensor powers of (globally) list-recoverable codes are 'approximately' locally list-recoverable, and that the 'approximately' modifier may be removed by pre-encoding the message with a suitable locally decodable code. Instantiating this with known constructions of high-rate globally list-recoverable codes and high-rate locally decodable codes finishes the construction.

Average-radius list-recoverability of random linear codes

“…A similar exponential hit in the list size is also present in [HRW17]. Our approach is able to establish better listrecovery of high-rate random linear codes.…”

“…In [HW15], the authors did use an off-the-shelf result for random linear codes, and this resulted in an exponential increase in their output list size. A similar blowup in the list size (because of the same reason) also happened in [HRW17].…”

“…Using our results from the previous bullet, one can improve the list size bounds in previous works that use list recovery of random linear codes as a building block [HW15,HRW17]. For example, this improves the list-size in the near-linear time list decodable codes (with optimal rate and block length n) of [HRW17] from exp (exp (exp (O( · log * n)))) to exp (exp (exp (log · log * n + O(1)))) for some large enough constant .…”

“…For example, this improves the list-size in the near-linear time list decodable codes (with optimal rate and block length n) of [HRW17] from exp (exp (exp (O( · log * n)))) to exp (exp (exp (log · log * n + O(1)))) for some large enough constant .…”

“…For example, in coding theory, listdecodability and list-recoverability of random linear codes is useful as a building block for other codingtheoretic constructions [GI01,HW15,HRW17], and ideas from these explorations have been useful in answering other coding-theoretic questions, for example the list-decodability of Reed-Solomon codes [RW14,RW15]. In complexity theory, list-decodability and related notions are tied to notions in pseudorandomness like extractors, expanders, and pseudorandom generators [Vad11]; in cryptography the task of algorithmically list-decoding random linear codes is related to learn- * atri@buffalo.edu.…”