Constructing integral matrices with given line sums

D. Gale, in 1957 and H.J. Ryser, in 1963, independently proved the famous Gale–Ryser theorem on the existence of (0, 1)–matrices with prescribed row and column sums. Around the same time, in 1968, Mirsky solved the more general problem of finding conditions for the existence of a nonnegative integral matrix with entries less than or equal to p and prescribed row and column sums. Using the results of Mirsky, Brualdi shows that a modified version of the domination condition of Gale–Ryser is still necessary and sufficient for the existence of a matrix under the same constraints. In this article we prove another extension of Gale–Ryser’s domination condition. Furthermore we present a method to build nonnegative integral matrices with entries less than or equal to p and prescribed row and column sums.