LRCs: Duality, LP Bounds, and Field Size

TL;DR

This paper develops a duality theory for locally recoverable codes (LRCs), deriving new bounds on their parameters and non-existence results over small fields, advancing the understanding of LRC limitations.

Contribution

It introduces a duality framework for LRCs, leading to improved bounds and non-existence results, especially over small fields.

Findings

Derived LP-type bounds that improve existing limits

Established non-existence of optimal LRCs over small fields

Linked code parameters to field size constraints

Abstract

We develop a duality theory of locally recoverable codes (LRCs) and apply it to establish a series of new bounds on their parameters. We introduce and study a refined notion of weight distribution that captures the code's locality. Using a duality result analogous to a MacWilliams identity, we then derive an LP-type bound that improves on the best known bounds in several instances. Using a dual distance bound and the theory of generalized weights, we obtain non-existence results for optimal LRCs over small fields. In particular, we show that an optimal LRC must have both minimum distance and block length relatively small compared to the field size.