Multidimensional matrices uniquely recovered by their lines (Q1938699)

A binary matrix is a lonesum if it can be uniquely recovered by its rows and columns sums. \textit{H. K. Kim}, \textit{D. S. Krotov} and the author [``Poly-Bernoulli numbers and lonesum matrices'', \url{arXiv:1103.4884}] extended several results on such matrices to \(q\)-ary matrices, i.e., to matrices whose entries are in \(\{0,1,\dots,q-1\}\). In this paper, the author provides two possible generalizations of \(q\)-ary lonesum matrices. Methods to determine whether a \(q\)-ary multidimensional matrix is a lonesum are discussed.