On Higher Dimensional Perfect Factors

A \(d\)-dimensional Perfect Factor is a collection of periodic arrays in which every \(k\)-ary \((n_1, \ldots, n_d)\) matrix appears exactly once (periodically). The one-dimensional case, with a collection of size one, is known as a De Bruijn cycle. The \(1\)- and \(2\)-dimensional versions have proven highly applicable in areas such as coding, communications, and location sensing. Here we focus on results in higher dimensions for factors with each \(n_i = 2\).