Codes for Graph Erasures

Abstract

Motivated by systems, where the information is represented by a graph such as neural networks, associative memories, and distributed systems. In this paper, we present a new class of codes, called codes over graphs . Under this paradigm, the information is stored on the edges of undirected or directed complete graphs, and a code over graphs is a set of graphs. A node failure is the event, where all edges in the neighborhood of the erased node have been erased. We say that a code over graphs can tolerate $\rho $ node failures, if it can correct the erased edges of any $\rho $ failed nodes in the graph. While the construction of optimal codes over graphs can be easily accomplished by MDS codes, their field size has to be at least ${\mathcal{ O}}(n^{2})$ , when $n$ is the number of nodes in the graph. In this paper, we present several constructions of codes over graphs with smaller field size. To accomplish this task, we use constructions of product codes and rank metric codes. Furthermore, we present optimal codes over graphs correcting two node failures over the binary field, when the number of nodes in the graph is a prime number. Last, we also provide upper bound on the number of nodes for optimal codes.