Existence of square-shaped de Bruijn tori for odd pattern sizes with even alphabet size

Determine whether square-shaped de Bruijn tori of type (M, M; n, n)k exist for odd pattern sizes n in {3,5,7} when the alphabet size k is even. Equivalently, ascertain the existence of de Bruijn tori with M = N and pattern shape (n, n) over an even-sized alphabet for n ∈ {3,5,7}.

Background

De Bruijn tori (perfect maps) require that every (m, n)-pattern over an alphabet of size k appears exactly once within one period. Necessary conditions include M ≥ m, N ≥ n, and MN = k^{mn}. For square-shaped tori with pattern shape (n, n), this implies M = N.

While many constructions are known, certain small-parameter cases remain unsettled. The abstract highlights as an example that square-shaped de Bruijn tori for odd n in {3,5,7} with even alphabet sizes are not known to exist, motivating the study of sub-perfect maps such as de Bruijn rings presented in the paper.

The paper later discusses related known results and conjectures (e.g., Hurlbert–Isaak) that characterize when such square-shaped tori exist, except for a set of small odd n where the problem remains unresolved.