# Local List Recovery of High-rate Tensor Codes and Applications

## TL;DR

This paper introduces the first high-rate locally list-recoverable codes, enabling capacity-achieving locally list-decodable and globally list-decodable codes with efficient decoding, advancing coding theory applications.

## Contribution

It presents the first construction of high-rate locally list-recoverable codes, achieving capacity and efficient decoding over constant-sized alphabets.

## Key findings

- First capacity-achieving locally list-decodable codes.
- First capacity-achieving globally list-decodable codes with near-linear time decoding.
- Binary codes on the Gilbert-Varshamov bound with near-linear-time unique decoding.

## Abstract

In this work, we give the first construction of high-rate locally list-recoverable codes. List-recovery 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.