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It is known that every self-dual binary code which is not doubly even is a "child" of a doubly even parent. It will be shown that an child of an doubly even parent has covering radius . Every extremal doubly even code has covering radius and every extremal doubly even code has covering radius . The complete coset weight distribution of the quadratic residue code is given, as well as bounds or exact values for the covering radii of all extremai doubly even codes of length less than or equal to .