On Extremal Rates of Storage Over Graphs

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.