Criteria of valid line sum arrays for multidimensional matrices

TL;DR

This paper extends Gale and Ryser's binary matrix existence criterion to multidimensional q-ary matrices, providing a new necessary and sufficient condition based on line sums.

Contribution

It introduces a higher-dimensional generalization of matrices and establishes a criterion for the existence of q-ary multidimensional matrices with specified line sums.

Findings

Established a criterion for multidimensional q-ary matrices with given line sums.

Generalized Gale-Ryser and Mirsky theorems to higher dimensions.

Provided a theoretical foundation for multidimensional matrix existence problems.

Abstract

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.