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This paper is concerned with the bifurcation of limit cycles in generalquadratic perturbations of quadratic codimension-four centers $Q_4$. Gavrilovand Iliev set an upper bound of {\it eight} for the number of limit cyclesproduced from the period annulus around the center. Based on Gavrilov-Iliev'sproof, we prove in this paper that the perturbed system has at most five limitcycles which emerge from the period annulus around the center. We also showthat there exists a perturbed system with three limit cycles produced by theperiod annulus of $Q_4$.

On the number of limit cycles in quadratic perturbations of quadratic codimension four centers