Des. Codes Cryptogr. (2011) 60:155–169 DOI 10.1007/s10623-010-9423-7 A recursive construction of nonbinary de Bruijn sequences Abbas Alhakim · Mufutau Akinwande Received: 28 December 2008 / Revised: 21 July 2010 / Accepted: 24 July 2010 / Published online: 19 August 2010 © Springer Science+Business Media, LLC 2010 Abstract This paper presents a method to find new de Bruijn sequences based on ones of lesser order. This is done by mapping a de Bruijn cycle to several vertex disjoint cycles in a de Bruijn digraph of higher order and then connecting these cycles into one full cycle. We present precise formulae for the locations where those cycles can be rejoined into one full cycle. We obtain an exponentially large class of distinct de Bruijn cycles. This method generalizes the Lempel construction of binary de Bruijn sequences as well as its efficient implementation by Annextein. Keywords De Bruijn sequence · Recursive construction · Graph homomorphism · Lempel’s D-morphism Mathematics Subject Classification (2000) 68R01 · 68R10 · 05C38 1 Introduction The Lempel homomorphism between binary de Bruijn digraphs of consecutive orders has been used by many authors to construct new de Bruijn sequences using a given one of smaller order as in [1, 3, 6], or to obtain results about the linear complexity of binary sequences as, e.g., in [2, 4] and references therein. Ronse gives an attempt in [7] to generalize Lempel’s construction by presenting a single function between two non-binary de Bruijn digraphs of consecutive orders that has a sim- Communicated by L. Teirlinck. A. Alhakim (B ) Department of Mathematics, American University of Beirut, Beirut, Lebanon e-mail: aa145@aub.edu.lb M. Akinwande Department of Mathematics and Computer Science, Clarkson University, Potsdam, NY 13699, USA e-mail: akinwamb@clarkson.edu 123