Duality and LP Bounds for Codes with Locality

TL;DR

This paper develops a duality theory for locally recoverable codes, introducing new invariants and bounds that enhance understanding and improve parameter limits compared to existing results.

Contribution

It characterizes code locality via dual codes, introduces refined invariants, and establishes a novel duality theorem leading to improved bounds for locally recoverable codes.

Findings

Established a duality theorem for locally recoverable codes.

Derived two new bounds, including an LP bound surpassing previous limits.

Enhanced understanding of code locality through dual code characterization.

Abstract

We initiate the study of the duality theory of locally recoverable codes, with a focus on the applications. We characterize the locality of a code in terms of the dual code, and introduce a class of invariants that refine the classical weight distribution. In this context, we establish a duality theorem analogous to (but very different from) a MacWilliams identity. As an application of our results, we obtain two new bounds for the parameters of a locally recoverable code, including an LP bound that improves on the best available bounds in several instances.