Say you have $m$ boolean inputs, and you are given a threshold $n$. You need to construct a boolean circuit that evaluates to true if at least $n$ of the inputs true. You may use AND, OR, NOT, or XOR gates (restricted to fan-in two, with arbitrary fan-out). Asymptotically how small can you make this circuit?
Any reasonably tight upper bound would be appreciated. I keep on thinking of ways to recursively construct such a circuit but I can't find anything good. Also, any results for any other reasonable basis of allowed gates would also be useful.