Constructing Orthogonal de Bruijn Sequences

Abstract. A(σ, k)-de Bruijn sequence is a minimum length string on an alphabet set of size σ which contains all σ k k-mers exactly once. Motivated by an application in synthetic biology, we say a given collection of de Bruijn sequences are orthogonal if no two of them contain the same (k + 1)-mer; that is, the length of their longest common substring is k. In this paper, we show how to construct large collections of orthogonal de Bruijn sequences. In particular, we prove that there are at least ⌊σ/2 ⌋ mutually-orthogonal order-k de Bruijn sequences on alphabets of size σ for all k. Based on this approach, we present a heuristic which proves capable of efficiently constructing optimal collections of mutuallyorthogonal sequences for small values of σ and k, which supports our conjecture that σ − 1 mutually-orthogonal de Bruijn sequences exist for all σ and k