Average-radius list-recovery of random linear codes: it really ties the room together

TL;DR

This paper introduces a new approach to analyze the list-decodability and list-recovery of random linear codes, unifying and extending previous results across various parameter regimes and code rates.

Contribution

It presents a novel method that applies broadly to list-recovery, improving understanding and bounds for random linear codes in multiple regimes.

Findings

Better list-decoding results for low-rate codes over large fields

Enhanced list-recovery bounds for high-rate codes

Reproduction of Guruswami-Hastad-Kopparty rate bounds with large list sizes

Abstract

We analyze the list-decodability, and related notions, of random linear codes. This has been studied extensively before: there are many different parameter regimes and many different variants. Previous works have used complementary styles of arguments---which each work in their own parameter regimes but not in others---and moreover have left some gaps in our understanding of the list-decodability of random linear codes. In particular, none of these arguments work well for list-recovery, a generalization of list-decoding that has been useful in a variety of settings. In this work, we present a new approach, which works across parameter regimes and further generalizes to list-recovery. Our main theorem can establish better list-decoding and list-recovery results for low-rate random linear codes over large fields; list-recovery of high-rate random linear codes; and it can recover the rate bounds of Guruswami, Hastad, and Kopparty for constant-rate random linear codes (although with large list sizes).