The simplified weighted sum function and its average sensitivity

• We simplify the definition of the weighted sum Boolean function.
• The new function is shown to have similar properties as the previous one.
• We give a tight bound on the average sensitivity of this new function.
• Numerical results show the Shparlinski's conjecture on the Fourier coefficients may not be true.

In this paper we simplify the definition of the weighted sum Boolean function which used to be inconvenient to compute and use. We show that the new function has essentially the same properties as the previous one. In particular, the bound on the average sensitivity of the weighted sum Boolean function remains unchanged after the simplification.