On Extremal Rates of Storage over Graphs

A storage code over a graph maps K independent source symbols, each of Lw bits, to N coded symbols, each of Lv 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 Lw/Lv 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.