Highly Nonlinear Balanced Boolean Functions with Very Good Autocorrelation Property

Constructing highly nonlinear balanced Boolean functions with very good autocorrelation property is an interesting open question. In this direction we use the measure Δf, the highest magnitude of all autocorrelation coefficients for a function f. We provide balanced functions f with currently best known nonlinearity and Δf values together. Our results for 15-variable functions disprove the conjecture proposed by Zhang and Zheng (1995) for different ranges of nonlinearity, where our constructions are based on modifications of Patterson-Wiedemann (1983) functions. Also we propose a simple bent based construction technique to get functions with very good Δf values for odd number of variables. This construction has a root in Kerdock Codes. Moreover, our construction on even number of variables is a recursive one and we conjecture (similar to Dobbertinʹs conjecture (1994) with respect to nonlinearity) that this provides the minimum possible value of Δf for a balanced function f on even number of variables.