De Bruijn Sequences for the Binary Strings with Maximum Density

Abstract

A de Bruijn sequence is a circular binary string of length 2ⁿ that contains each binary string of length n exactly once as a substring. A maximum-density de Bruijn sequence is a circular binary string of length $\binom{n}{0} + \binom{n}{1} + \binom{n}{2} + \cdots + \binom{n}{m}$ that contains each binary string of length n with density (number of 1s) between 0 and m, inclusively. In this paper we efficiently generate maximum-density de Bruijn sequences for all values of n and m. An interesting special case occurs when n = 2m + 1. In this case our result is a “complement-free de Bruijn sequence” since it is a circular binary string of length 2^n − 1 that contains each binary string of length n or its complement exactly once as a substring.