Distribution of linear and second-highest degree parts of maximum-length feedback functions

De Bruijn sequences are maximum-length sequences generated by feedback shift registers. The study of their feedback functions is very important and has attracted a lot of attention. Many necessary conditions are proposed so far. In this manuscript, based on the complementary and converse properties of de Bruijn sequences, and on some special cross-join pairs in de Bruijn sequences, we give several properties about the proportions of linear part and second-highest degree part of de Bruijn sequence feedback functions. We prove that at least \(\frac{3}{4}\) $\frac{3}{4}$ proportion de Bruijn sequence feedback functions of order n have their second-highest degree \(n-2\) $n-2$ . We also exhaust all possible Boolean functions and discover all feedback functions of order 6 de Bruijn sequences and feedback functions with second-highest degree 2, of order 7, 8 and 9 de Bruijn sequences. Then we statistically analyze these functions according to their linear parts, second-highest degrees and number of terms. The result shows that their distribution according to degree \(n-2\) $n-2$ parts is almost uniform. But the distribution according to linear parts has significant bias, which means that certain kinds of feedback functions are more likely to generate de Bruijn sequences.