De Bruijn Sequences for Fixed-Weight Binary Strings

Abstract

De Bruijn sequences are circular strings of length $2^n$ whose length n substrings are the binary strings of length n. Our focus is on creating circular strings of length $\binom{n}{w}$ for the binary strings of length n with weight (number of 1s) equal to w. In this case, each fixed-weight string can be encoded by its first $n{-}1$ bits since the final bit is redundant. For this reason, we construct circular strings of length $\binom{n-1}{w}+\binom{n-1}{w-1}$ whose length $n{-}1$ substrings are the binary strings of length $n{-}1$ with weight w or $w{-}1$. Our construction is reminiscent of the construction for the lexicographically least de Bruijn sequence, except the underlying algorithm is applied to cool-lex order instead of lexicographic order. The construction can be efficiently implemented so that successive blocks of n bits are generated in constant amortized time while using $O(n \log n)$-space. This article's results were also used to create de Bruijn sequences for binary strings of length n with a specified maximum weight.