Efficient generation of odd order de Bruijn sequence with the same complement and reverse sequences

TL;DR

This paper presents an efficient method for generating de Bruijn sequences of odd order where the complement and reverse sequences are identical, solving a long-standing open problem and analyzing their properties.

Contribution

It introduces a novel efficient generation method for specific de Bruijn sequences and refines their characterization, addressing a problem posed over forty years ago.

Findings

Existence of de Bruijn sequences with same complement and reverse for odd n

An efficient generation method for these sequences

Analysis of their number and linear complexity distribution

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.