M. B. Levin constructed a real number x such that the first N terms of the sequence b^n x mod 1 for n >= 1 have discrepancy $O((log N)^2/N)$.  This is the lowest discrepancy known for this kind of sequences.  In this talk, we present Levin's construction in terms of nested perfect necklaces, which are a variant of the classical de Bruijn sequences.  For base 2 and the order being a power of 2, we give the exact number of nested perfect necklaces and an explicit method based on matrices to construct each of them.