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The "defect" of a curve over a finite field is the difference between thenumber of rational points on the curve and the Weil-Serre bound for the curve.We present a construction for producing genus-4 double covers of genus-2 curvesover finite fields such that the defect of the double cover is not much morethan the defect of the genus-2 curve. We give an algorithm that uses thisconstruction to produce genus-4 curves with small defect. Heuristically, forall sufficiently large primes and for almost all prime powers q, the algorithmis expected to produce a genus-4 curve over F_q with defect at most 4 in timeq^{3/4}, up to logarithmic factors.  As part of the analysis of the algorithm, we present a reinterpretation ofresults of Hayashida on the number of genus-2 curves whose Jacobians areisomorphic to the square of a given elliptic curve with complex multiplicationby a maximal order. We show that a category of principal polarizations on thesquare of such an elliptic curve is equivalent to a category of right ideals ina certain quaternion order.

Quickly constructing curves of genus 4 with many points