Perceptrons of large weight

A threshold gate is a linear combination of input variables with integer coefficients (weights). It outputs 1 if the sum is positive. The maximum absolute value of the coefficients of a threshold gate is called its weight. A degree-$d$ perceptron is a Boolean circuit of depth 2 with a threshold gate at the top and any Boolean elements of fan-in at most $d$ at the bottom level. The weight of a perceptron is the weight of its threshold gate.
For any constant $d\ge 2$ independent of the number of input variables $n$, we construct a degree-$d$ perceptron that requires weights of at least $n^{\Omega(n^d)}$; i.e., the weight of any degree-$d$ perceptron that computes the same Boolean function must be at least $n^{\Omega(n^d)}$. This bound is tight: any degree-$d$ perceptron is equivalent to a degree-$d$ perceptron of weight $n^{O(n^d)}$. For the case of threshold gates (i.e., $d=1$), the result was proved by Håstad in [2]; we use Håstad's technique.