The cycle structure of a binary linear shift register with connection polynomial <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G(x)=(1+x)^{2}g(x)</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g(x)</tex> is a primitive polynomial of degree <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m-2</tex> over GF <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(2)</tex> , is used to give several construction techniques for generation of shift-register sequences of length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l=2^{m}-4</tex> . It is shown that a class of nonlinear deBruijn cycles, where the number of elements is proportional to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2^{5m}</tex> , can be constructed. The obtained cycles can be generated by simple <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> -stage nonlinear feedback shift registers.