Enumeration of Both-Ends-Fixed <i>k</i>-Ary Necklaces and Its Applications

Abstract

We consider both-ends-fixed k-ary necklaces and enumerate all such necklaces of length n from the viewpoints of symbolic dynamics and β-expansions, where n and k(≥ 2) are natural numbers and β(> 1) is a real number. Recently, Sawada et al. proposed an efficient construction of k-ary de Bruijn sequence of length kn, which for each n ≥ 1, requires O(n) space but generates a single k-ary de Bruijn sequence of length kn in O(1)-amortized time per bit. Based on the enumeration of both-ends-fixed k-ary necklaces of length n, we evaluate auto-correlation values of the k-ary de Bruijn sequences of length kn constructed by Sawada et al. We also estimate the asymptotic behaviour of the obtained auto-correlation values as n tends to infinity.