On weights in duadic codes

AbstractIn [3] we introduced a new family of binary, cyclic (n, (n+1)2) and (n, (n-1)2) codes which include quadratic residue (Q. R.) codes when n is prime. These codes are defined in terms of their idempotent generators and they exist for all n=p1a1p2a2…prar where each pi is a prime ≡ ±1 (mod 8). This family of codes has properties analogous to properties of Q.R. codes, and we investigate some of these in this paper. In particular, we determine conditions on n under which one or every extended duadic code is self-dual or its dual is the other associated duadic code. In the self-dual case we characterize weights and strengthen the square root bound. In addition, we show that every extended cyclic, self-dual binary code is a duadic code. Hence we know at which lengths these codes exist and that certain Reed-Muller codes are duadic codes. In (Leon, Masley, and Pless, IEEE. Trans. Inform. Theory IT 30 (1984), 709–714) information was given about duadic codes until length 119. Here we give extensive tables describing minimum weights and duals of all duadic codes until length 241. Several duadic codes are better than the Q.R. codes of the same length, and one is the best code known of its length