Orientable manifolds with nonzero dual Stiefel-Whitney classes of largest possible grading

It is known that, for all n , there exist compact differentiable orientable n -manifolds with dual Stiefel-Whitney class w ‾ n − α ˆ ( n ) ≠ 0 , and this is best possible, but the proof is nonconstructive. Here α ˆ ( n ) equals the number of 1's in the binary expansion of n if n ≡ 1 mod 4, and exceeds this by 1 otherwise. We find, for all n ≢ 0 mod 4, examples of real Bott maanifolds with this property.