Abstract
A new family of binary cyclic (n,(n + 1)/2) and (n,(n - 1)/2) codes are introduced, which include quadratic residue (QR) codes when n is prime. These codes are defined in terms of their idempotent generators, and they exist for all odd n = p_{1}^{a_{1}} p_{2}^{a_{2}} \cdots p_{r}^{a_{r}} where each p_{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 than 127 are identified, and the minimum weights of their extensions are given. One of the duadic codes of length 113 has greater minimum weight than the QR code of that length.
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