# Locally Repairable Codes with Unequal Local Erasure Correction

## TL;DR

This paper investigates the tradeoff between locality and minimum distance in locally repairable codes with arbitrary local code requirements, providing bounds and optimal constructions for non-homogeneous distributed storage systems.

## Contribution

It introduces a new framework allowing variable local code requirements, derives Singleton-type bounds, and constructs optimal codes for these settings.

## Key findings

- Derived Singleton-type distance bounds for variable local code requirements.
- Provided an optimal code construction matching the bounds.
- Characterized the feasible rate region independent of local distance constraints.

## Abstract

When a node in a distributed storage system fails, it needs to be promptly repaired to maintain system integrity. While typical erasure codes can provide a significant storage advantage over replication, they suffer from poor repair efficiency. Locally repairable codes (LRCs) tackle this issue by reducing the number of nodes participating in the repair process (locality), at the cost of reduced minimum distance. In this paper, we study the tradeoff between locality and minimum distance of LRCs with local codes that have arbitrary distance requirements. Unlike existing methods where both the locality and the local distance requirements imposed on every node are identical, we allow the requirements to vary arbitrarily from node to node. Such a property can be an advantage for distributed storage systems with non-homogeneous characteristics. We present Singleton-type distance upper bounds and also provide an optimal code construction with respect to these bounds. In addition, the feasible rate region is characterized by dimension upper bounds that do not depend on the distance.