Criteria of valid line sum arrays for multidimensional matrices

Gale and Ryser found a criterion for existence of a binary matrix with given row and column sums. Mirsky extended the theorem of Gale and Ryser to $q$-ary matrices. In this paper, we are interested in higher dimensional extension of these theorems. We first introduce multidimensional matrices as a higher dimensional generalization of matrices. We next replace the concept of row and columns in matrices by lines in multidimensional matrices. We finally find a criterion for existence of a $q$-ary multidimensional matrices with given line sums.