The Coolest Order of Binary Strings

Abstract

Pick a binary string of length n and remove its first bit b. Now insert b after the first remaining 10, or insert $\overline{b}$ at the end if there is no remaining 10. Do it again. And again. Keep going! Eventually, you will cycle through all 2ⁿ of the binary strings of length n. For example, are the binary strings of length n = 4, where 1 = and 0 = Che bello! And if you only want strings with weight (number of 1s) between ℓ and u? Just insert b instead of $\overline{b}$ when the result would have too many 1s or too few 1s. For example, are the strings with n = 4, ℓ = 0 and u = 2. Strabello! This generalizes ‘cool-lex’ order by Ruskey and Williams (The coolest way to generate combinations, Discrete Mathematics). We use it to construct de Bruijn sequences for (i) ℓ = 0 and any u (maximum specified weight), (ii) any ℓ and u = n (minimum specified weight), and (iii) odd u − ℓ (even size weight range). For example, all binary strings with n = 6, ℓ = 1, and u = 4 appear once (cyclically) in