Studying the case of de Bruijn parity vectors is questionable: They are easily ruled out as cycle candidates due to the trivial property that, for those vectors, l(v) = 2s(v). Hence, it can be shown that the smallest term is lower than 1 and cannot be an integer, unless we consider the trivial cyle whose parity vector 10 is the smallest de Bruijn sequence.
A more relevant collection to study might be the set V of those vectors v such that
- s(v) = \lfloor (\log_3 2) \, l(v)\rfloor, so that \delta is minimal (and positive)
- LRS(v) = k
- there is no vector longer than v holding the two above properties (for maximality)
where k is a fixed integer.
For example, when k=2, we get two vectors of length 7 and weight 4
1110100
1110010
and their rotations.
In a sense, it is a generalization of de Bruijn sequences with an extra constraint on the ones-ratio s(v)/l(v), which has probably been studied already. References are welcome.