Randall Dougherty

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It is known that every solvable multicast network has a scalar linear solution over a sufficiently large finite-field alphabet. It is also known that this result does not generalize to arbitrary networks. There are several examples in the literature of solvable networks with no scalar linear solution over any finite field. However, each example has a linear(More)
We define a class of networks, called matroidal networks, which includes as special cases all scalar-linearly solvable networks, and in particular solvable multicast networks. We then present a method for constructing matroidal networks from known matroids. We specifically construct networks that play an important role in proving results in the literature,(More)
Any unconstrained information inequality in three or fewer random variables can be written as a linear combination of instances of Shannon’s inequality I(A;B|C) ≥ 0. Such inequalities are sometimes referred to as “Shannon” inequalities. In 1998, Zhang and Yeung gave the first example of a “nonShannon” information inequality in four variables. Their(More)
We define the routing capacity of a network to be the supremum of all possible fractional message throughputs achievable by routing. We prove that the routing capacity of every network is achievable and rational, we present an algorithm for its computation, and we prove that every rational number in (0, 1] is the routing capacity of some solvable network.(More)
We prove that for any finite directed acyclic network, there exists a corresponding multiple-unicast network, such that for every alphabet, each network is solvable if and only if the other is solvable, and, for every finite-field alphabet, each network is linearly solvable if and only if the other is linearly solvable. The proof is constructive and creates(More)
Ranks of subspaces of vector spaces satisfy all linear inequalities satisfied by entropies (including the standard Shannon inequalities) and an additional inequality due to Ingleton. It is known that the Shannon and Ingleton inequalities generate all such linear rank inequalities on up to four variables, but it has been an open question whether additional(More)
If beta and gamma are nonnegative integers and F is a field, then a polynomial collection {p<sub>1</sub>,<sub>hellip</sub> ,P<sub>beta</sub>} sube Z[alpha<sub>1,hellip</sub>, alpha<sub>gamma</sub>] is said to be solvable over F if there exist omega<sub>1hellip</sub>, omega<sub>gamma</sub> isin F such that for all i = 1,<sub>hellip</sub>, beta we have(More)
The coding capacity of a network is the supremum of ratios <i>k/n</i> for which there exists a fractional (<i>k, n</i>) coding solution, where <i>k</i> is the source message dimension and <i>n</i> is the maximum edge dimension. The coding capacity is referred to as routing capacity in the case when only routing is allowed. A network is said to achieve its(More)
We address the degree-diameter problem for Cayley graphs of Abelian groups (Abelian graphs), both directed and undirected. The problem turns out to be closely related to the problem of finding efficient lattice coverings of Euclidean space by shapes such as octahedra and tetrahedra; we exploit this relationship in both directions. For 2 generators(More)