The Generalized Covering Radii of Linear Codes

TL;DR

This paper introduces the generalized covering radius of linear codes, linking it to database query efficiency, and explores its properties, bounds, and connections with other code parameters.

Contribution

It defines and analyzes the generalized covering radius of linear codes, providing bounds and establishing connections with generalized Hamming weights.

Findings

Derived bounds on code parameters related to generalized covering radii.

Established equivalences of the generalized covering radius as combinatorial, geometric, and algebraic.

Explored the impact of code operations on the generalized covering radius.

Abstract

Motivated by an application to database linear querying, such as private information-retrieval protocols, we suggest a fundamental property of linear codes -- the generalized covering radius. The generalized covering-radius hierarchy of a linear code characterizes the trade-off between storage amount, latency, and access complexity, in such database systems. Several equivalent definitions are provided, showing this as a combinatorial, geometric, and algebraic notion. We derive bounds on the code parameters in relation with the generalized covering radii, study the effect of simple code operations, and describe a connection with generalized Hamming weights.