Proofs of Conjectures on Extremal Weight De Bruijn Sequences

De Bruijn sequences can be categorized by the weight of the truth tables of the generating functions. Among all the weight classes, the numbers of de Bruijn sequences with minimum and maximum weights draw much attention. Fredricksen and Mayhew proposed six conjectures about them, some of which have remained unsolved for about forty years. In this paper, we give the exact formulas of de Bruijn sequences with extremal weights and then prove all the conjectures in the affirmative.