A simple shift rule for k-ary de Bruijn sequences

A k-ary de Bruijn sequence of order n is a cyclic sequence of length kⁿ in which each k-ary string of length n appears exactly once as a substring. A shift rule for a de Bruijn sequence of order n is a function that maps each length n substring to the next length n substring in the sequence. We present the first known shift rule for k-ary de Bruijn sequences that runs in O(1)-amortized time per symbol using O(n) space. Our rule generalizes the authors’ recent shift rule for the binary case (A surprisingly simple de Bruijn sequence construction, Discrete Math. 339, 127–131).