Storage codes and recoverable systems on lines and grids

TL;DR

This paper introduces a new method for constructing storage codes using resolvable designs and analyzes their capacity on lines and grids, revealing explicit formulas for various one- and two-dimensional systems.

Contribution

It presents a novel construction technique for storage codes via interleaving with resolvable designs and derives capacity formulas for codes on lines and grids.

Findings

New construction method for storage codes using resolvable designs

Closed-form capacity expressions for codes on lines and grids

Connections established between storage codes, graphs, and difference-avoiding sets

Abstract

A storage code is an assignment of symbols to the vertices of a connected graph $G(V,E)$ with the property that the value of each vertex is a function of the values of its neighbors, or more generally, of a certain neighborhood of the vertex in $G$. In this work we introduce a new construction method of storage codes, enabling one to construct new codes from known ones via an interleaving procedure driven by resolvable designs. We also study storage codes on $\mathbb Z$ and ${\mathbb Z}^2$ (lines and grids), finding closed-form expressions for the capacity of several one and two-dimensional systems depending on their recovery set, using connections between storage codes, graphs, anticodes, and difference-avoiding sets.