A linear threshold element computes a function that is a sign of a weighted sum of the
input variables. The weights are arbitrary integers; actually, they can be very big integers-
exponential in the number of the input variables. While in the present literature a distinction is
made between the two extreme cases of linear threshold functions with polynomial-size weights
as opposed to those with exponential-size weights, the best known lower bounds on the size
of threshold circuits are for depth-2 circuits with small weights. Our main contributions are
devising two distinct methods for constructing threshold functions with minimal weights and
filling up the gap between polynomial and exponential weight growth by further refining the
separation. Namely, we prove that the class of linear threshold functions with polynomial-size
weights can be divided into subclasses according to the degree of the polynomial. In fact, we
prove a more general result-that there exists a minimal weight linear threshold function for
any arbitrary number of inputs and any weight size
