cient Digital to Analog Encoding

An important issue in analog circuit design is the problem of digital to analog conversion, namely, the encoding of Boolean variables into a single analog value which contains enough information to reconstruct the values of the Boolean variables. A natural question is: What is the complexity of implementing the digital to analog encoding function? That question was recently answered in 5], where matching lower and upper bounds on the size of the circuit for the encoding function were proven. In particular, it was proven that 3n?1 2 2-input arithmetic gates are necessary and suucient for implementing the encoding function of n Boolean variables. However, the proof of the upper bound is not constructive. In this paper, we present an explicit construction of a digital to analog encoder that is optimal in the number of 2-input arithmetic gates. In addition, we present an eecient analog to digital decoding algorithm. Namely, given the encoded analog value, our decoding algorithm reconstructs the original Boolean values. Our construction is suboptimal in that it uses constants of maximum size n log n bits while the non-constructive proof uses constants of maximum size 2n + dlog ne bits.