An algorithm for generating all CR sequences in the de Bruijn sequences of length 2ⁿ where n is any odd number

Abstract

For the case that p is any prime number, we have already constructed all CR (complement reverse) sequences in the de Bruijn sequences of length 2^2p+1. In this research, with the help of the Dyck language, we characterize CR sequences in the de Bruijn sequences of length 2^2m+1 where m (≥4) is a non-prime number. In virtue of this characterization, we show that for any odd number n, there exist CR sequences in the de Bruijn sequences of length 2ⁿ, which completely settles the fundamental problem posed by Fredricksen on existence of the CR sequences. Consequently, we establish an algorithm for generating all CR sequences in the de Bruijn sequences of length 2ⁿ for any odd n.