On semi-perfect de Bruijn words

We show an application of Lempel's recursive construction of De Bruijn words to the generation of binary words having many factor-rich prefixes. A binary word is said to be factor-rich iff it has the largest number of distinct factors among binary words with the same length. A linear de Bruijn word of rank n is a shortest word containing (as a factor) exactly once each binary word of length n. It is factor-rich and its length equals Δn=2n+n−1. We construct for each n a binary linear de Bruijn word of rank n which is semi-perfect in the following sense: each of its prefixes of length m>Δn−1 is factor-rich. The number Δn−1 is the best possible (for n>2 there is no linear binary de Bruijn word with factor-rich prefix of length m=Δn−1). We show an efficient algorithm constructing compact description of binary semi-perfect de Bruijn words.