On the Generalized Covering Radii of Reed-Muller Codes

TL;DR

This paper investigates the generalized covering radii of Reed-Muller codes, establishing bounds, exact values in special cases, and providing a polynomial-time covering algorithm with applications to data-query protocols.

Contribution

It offers new bounds and exact values for the generalized covering radii of Reed-Muller codes and introduces an efficient covering algorithm for use in data-query protocols.

Findings

Derived bounds on generalized covering radii

Exact values in certain extreme cases

Polynomial-time covering algorithm

Abstract

We study generalized covering radii, a fundamental property of linear codes that characterizes the trade-off between storage, latency, and access in linear data-query protocols such as PIR. We prove lower and upper bounds on the generalized covering radii of Reed-Muller codes, as well as finding their exact value in certain extreme cases. With the application to linear data-query protocols in mind, we also construct a covering algorithm that gets as input a set of points in space, and find a corresponding set of codewords from the Reed-Muller code that are jointly not farther away from the input than the upper bound on the generalized covering radius of the code. We prove that the algorithm runs in time that is polynomial in the code parameters.