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A new family of binary cyclic and codes are introduced, which include quadratic residue (QR) codes when is prime. These codes are defined in terms of their idempotent generators, and they exist for all odd where each is a prime . Dual codes are identified. The minimum odd weight of a duadic code satisfies a square root bound. When equality holds in the sharper form of this bound, vectors of minimum weight hold a projective plane. The unique projective plane of order 8 is held by the minimum weight vectors in two inequivalent duadic codes. All duadic codes of length less than are identified, and the minimum weights of their extensions are given. One of the duadic codes of length has greater minimum weight than the QR code of that length.