Local List Recovery of High-Rate Tensor Codes and Applications
Brett Hemenway, Noga Ron‐Zewi, Mary K. Wootters · SIAM Journal on Computing · 2019
We show that the tensor product of a high-rate globally list recoverable code is (approximately) locally list recoverable. List recovery has been a useful building block in the design of list decodable codes, and our motivation is to use the tensor construction as such a building block. In particular, instantiating this construction with known constructions of high-rate globally list recoverable codes, and using appropriate transformations, we obtain the first capacity-achieving locally list decodable codes (over a large constant size alphabet), and the first capacity-achieving globally list decodable codes with nearly linear time list decoding algorithms. Our techniques are inspired by an approach of Gopalan, Guruswami, and Raghavendra [ SIAM J. Comput., 40 (2011), pp. 1432--1462] for list decoding tensor codes.