Vector Linear Solution iff Dimension $\geq m$
Niladri Das, Brijesh Kumar Rai · IEEE Communications Letters · 2019
If a network has an m-dimensional vector linear network coding (VLNC) solution, then it also has a VLNC solution for all dimensions multiple of m; but can a VLNC solution exists for dimensions not a multiple of m? If m is the least positive integer such that a network has an m-dimensional VLNC solution, none of the networks shown in the literature has a VLNC solution for a dimension which is not a multiple of m. In this letter, we show that for any positive integer m ≥ 3, there exists a network which has no vector linear solution if the dimension is less than m - 1 but has a vector linear solution for all dimensions greater than or equal to m - 1.