A new family of binary cyclic(n,(n + 1)/2)and(n,(n - 1)/2)codes are introduced, which include quadratic residue (QR) codes whennis prime. These codes are defined in terms of their idempotent generators, and they exist for all oddn = p_{1}^{a_{1}} p_{2}^{a_{2}} \cdots p_{r}^{a_{r}}where eachp_{i}is a prime\equiv \pm 1 \pmod{8}. Dual codes are identified. The minimum odd weight of a duadic(n,(n + 1)/2)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(73,37,9)duadic codes. All duadic codes of length less than127are identified, and the minimum weights of their extensions are given. One of the duadic codes of length113has greater minimum weight than the QR code of that length.