Efficient generation of odd order de Bruijn sequence with the same complement and reverse sequences
Zuling Chang, Qiang Wang
TL;DR
The paper addresses Fredricksen’s question by proving that, for every odd order $n>1$, there exist de Bruijn sequences ${\mathbf s}$ with ${\mathbf c}{\mathbf s}= {\mathbf r}{\mathbf s}$ and provides a constructive CJM-based method using a PCR cycle decomposition by weight to produce many CR sequences. It introduces two companion successor rules that partition PCR cycles into low- and high-weight sets, form large cycles that satisfy $S_0=\mathbf{cr}S_1$, and merge them to yield CR de Bruijn sequences with an efficient $O(n)$-time, $O(n)$-space next-bit generation. The work further refines the enumeration and distribution of CR sequences, showing the count is a multiple of $2^{\frac{n+3}{2}}$ (and $2^{\frac{n+5}{2}}$ when accounting for a symmetry group) for odd $n\ge 5$, and demonstrates the existence of CR de Bruijn sequences with maximum linear complexity, with $\\gamma(2^n-1,n)\equiv 0 \pmod{16}$. These results advance understanding of de Bruijn sequence symmetry and provide practical, efficient generation of CR sequences for applications in coding, cryptography, and communications.
Abstract
Experimental results show that, when the order $n$ is odd, there are de Bruijn sequences such that the corresponding complement sequence and the reverse sequence are the same. In this paper, we propose one efficient method to generate such de Bruijn sequences. This solves an open problem asked by Fredricksen forty years ago for showing the existence of such de Bruijn sequences when the odd order $n >1$. Moreover, we refine a characterization of de Bruijn sequences with the same complement and reverse sequences and study the number of these de Bruijn sequences, as well as the distribution of de Bruijn sequences of the maximum linear complexity.