De Bruijn sequences for the binary strings with maximum density

Authors:
Joe Sawada;Brett Stevens;Aaron Williams
Affiliations:
School of Computer Science, University of Guelph, Canada;Department of Mathematics and Statistics, Carleton University, Canada;Department of Mathematics and Statistics, Carleton University, Canada
Venue:
WALCOM'11 Proceedings of the 5th international conference on WALCOM: algorithms and computation
Year:
2011

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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 n (n 0)+(n 1)+(n 2)+...+(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 2n-1 that contains each binary string of length n or its complement exactly once as a substring.