Codes for Erasures over Directed Graphs

TL;DR

This paper introduces a new class of codes over graphs designed for storage systems on directed graphs, focusing on constructing optimal binary codes capable of correcting two node failures with a prime number of nodes.

Contribution

The paper presents the first construction of optimal binary codes over graphs that can correct two node failures, reducing the required field size for such codes.

Findings

Constructed optimal binary codes over graphs for two node failures.

Achieved correction capability with a prime number of nodes.

Reduced field size requirements compared to previous codes.

Abstract

In this work we continue the study of a new class of codes, called \emph{codes over graphs}. Here we consider storage systems where the information is stored on the edges of a complete directed graph with $n$ nodes. The failure model we consider is of \emph{node failures} which are erasures of all edges, both incoming and outgoing, connected to the failed node. It is said that a code over graphs is a \textit{$\rho$-node-erasure-correcting code} if it can correct the failure of any $\rho$ nodes in the graphs of the code. While the construction of such optimal codes is an easy task if the field size is ${\cal O} (n^2)$, our main goal in the paper is the construction of codes over smaller fields. In particular, our main result is the construction of optimal binary codes over graphs which correct two node failures with a prime number of nodes.