On Higher Dimensional Perfect Factors

Abstract

A $d$-dimensional Perfect Factor is a collection of periodic arrays in which every $k$-ary $(n_1,\dots ,n_d)$ 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 $n_i=2$.