Stretching de Bruijn sequences

Abstract

We give a one-step construction of de Bruijn sequences of general alphabet size and with order \(n+k\), given a de Bruijn sequence of order n and any integer \(k>1\). This is achieved by using an appropriate class of graph homomorphisms between de Bruijn digraphs whose orders differ by an integer k. The method starts with a lower order de Bruijn cycle, finds its inverse cycles in the higher order digraph, which are then cross-joined into one full cycle. Therefore, this generalizes the Lempel’s binary construction and the Alhakim–Akinwande construction for non-binary alphabets and a wide class of homomorphisms.