A general theorem is proved showing how to obtain a constant-weight binary cyclic code from a p-ary linear cyclic code, where p is a prime, by using a representation of GF(p) as cyclic shifts of a binary p-tuple. Based on this theorem, constructions are given for four classes of binary constant-weight codes. The first two classes are shown to achieve the Johnson upper bound on minimum distance asymptotically for long block lengths. The other two classes are shown similarly to meet asymptotically the low-rate Plotkin upper bound on minimum distance. A simple method is given for selecting virtually the maximum number of cyclically distinct codewords with full cyclic order from Reed-Solomon codes and from Berlekamp-Justesen maximum-distance-separable codes. Two correspondingly optimum classes of constant-weight cyclically permutable codes are constructed. It is shown that cyclically permutable codes provide a natural solution to the problem of constructing protocol-sequence sets for the M-active-out-of-T-users collision channel without feedback.>