A generalized recursive construction for de Bruijn sequences

This paper is concerned with the construction of de Bruijn sequences of span
n
--binary sequences of period
2^{n}
in which every binary
n
-tuple appears as some
n
consecutive terms in one period of the sequence. Constructions in the literature are based on maximum length linear sequences, algorithms which start from scratch, and recursive methods which start with a single de Bruijn sequence of span
n
and produce one of span
n+1
. We give a more general recursive construction which takes two de Brnijn sequences of span
n
and produces a de Bruijn sequence of span
n+1
. In addition, for a special case of the construction, the complexity (shortest linear recursion that generates the sequence) of the resulting sequence is determined in terms of the complexities of the ingredient sequences. In particular, de Bruijn sequences of span
n+1
with maximum complexity
2^{(n+1)}-1
are obtained from maximum complexity sequences of span
n
. Reverse-complementary de Bruijn sequences are also considered.