Representing (0, 1)-matrices by boolean circuits

A boolean circuit represents an n by n(0,1)-matrix A if it correctly computes the linear transformation y->=Ax-> over GF(2) on all n unit vectors. If we only allow linear boolean functions as gates, then some matrices cannot be represented using fewer than @W(n^2/lnn) wires. We first show that using non-linear gates one can save a lot of wires: any matrix can be represented by a depth-2 circuit with O(nlnn) wires using multilinear polynomials over GF(2) of relatively small degree as gates. We then show that this cannot be substantially improved: If any two columns of an n by n(0,1)-matrix differ in at least d rows, then the matrix requires @W(dlnn/lnlnn) wires in any depth-2 circuit, even if arbitrary boolean functions are allowed as gates.

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