On Extremal Rates of Storage over Graphs

TL;DR

This paper investigates the maximum achievable symbol rate (capacity) of storage codes over graphs, establishing the top three capacity values and characterizing graphs that attain these capacities.

Contribution

It identifies the three highest storage capacity values over graphs and characterizes the graphs that achieve the top two capacities, providing conditions for the third.

Findings

Highest capacities are 2, 3/2, and 4/3.

Graphs with capacity 2 and 3/2 are fully characterized.

Necessary and sufficient conditions for capacity 4/3 are provided, but do not match.

Abstract

A storage code over a graph maps $K$ independent source symbols, each of $L_w$ bits, to $N$ coded symbols, each of $L_v$ bits, such that each coded symbol is stored in a node of the graph and each edge of the graph is associated with one source symbol. From a pair of nodes connected by an edge, the source symbol that is associated with the edge can be decoded. The ratio $L_w/L_v$ is called the symbol rate of a storage code and the highest symbol rate is called the capacity. We show that the three highest capacity values of storage codes over graphs are $2, 3/2, 4/3$. We characterize all graphs over which the storage code capacity is $2$ and $3/2$, and for capacity value of $4/3$, necessary condition and sufficient condition (that do not match) on the graphs are given.