Locality for Codes over the Integers

TL;DR

This paper investigates integer-based codes with locality constraints, introducing a weighted locality concept and deriving bounds, along with proposing new code constructions including integer analogs of Tamo--Barg codes.

Contribution

It introduces a weighted locality notion for codes over product rings, establishes a Singleton-like bound, and constructs new codes with locality, including integer analogs of known codes.

Findings

Derived a Singleton-like bound for locally recoverable codes over rings.

Proposed new code constructions with locality, including integer analogs of Tamo--Barg codes.

Extended the concept of locality to weighted notions over product rings.

Abstract

In this work, we study the codes over the integers with locality constraints. We introduce a weighted notion of locality over $\mathbb{Z}/q_1\mathbb{Z} \times \cdots \times \mathbb{Z}/q_n\mathbb{Z}$ and derive a Singleton-like bound for locally recoverable codes. We also propose some code constructions with locality, including integer analogs of Tamo--Barg codes.