Abstract
We present a simple explicit construction giving unbounded fan-in circuits with o( n) gates and depth O( r) for the threshold function of n variables when the threshold is at most ( log n) r , for any integer r > 0. This improves a result of Ajtai and Ben-Or, who showed the existence of circuits of size n o(1) . This is the highest threshold for which polynomial-size, constant-depth circuits are possible.
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