The dimension of magic squares over fields of characteristics two and three

Hou et al. [4] have studied various spaces of magic squares over a field F and determined their dimensions. However, they left one open question unsolved, namely, if the characteristic of F is 2 or 3, exactly which n and k make Mn,k(1) nonempty, where Mn,k(1) denotes the set of all n×n matrices over F whose row sums, column sums, k diagonal sums, and k antidiagonal sums are all 1. We solve this completely.