A construction of all CR sequences in the de Bruijn sequences of length 2^2p+1 where p is a prime number

Abstract

We partially solve a fundamental problem posed by Fredricksen on existence of CR (complement reverse) sequences in the de Bruijn sequences of length 2^2p+1(p≥1). For the case that p is a prime number, we construct the set of CR graphs, which yields all CR sequences. As an application of this result, we discuss enumeration of the total number of distinct auto-correlation functions for the set of de Bruijn sequences of length 2^2p+1. Since the worst cases of the normalized cross-correlation functions for pairs of de Bruijn sequences are characterized by the normalized auto-correlation functions for sequences of the worst pairs, we obtain upper bounds of the total number of distinct auto-correlation functions for the set of de Bruijn sequences of length 2ⁿ(n≥4).