The authors describe a method for producing Boolean functions of degree d3 in n=2dk-1 (k=1, 2, ...) variables, such that the functions are plateaued and balanced, have high nonlinearity and have no linear structures. The nonlinearity is 2(n-1)-2((n-1)/2), which is the same as the largest possible nonlinearity for a quadratic function in n (odd) variables (the so-called quadratic bound'). Their theorem uses some new ideas to generalise a theorem, which gave the case d=3, in a 2009 paper by Fengrong Zhang et al. They discuss the cryptographic properties and applications for the functions.