Constructions for optimal constant weight cyclically permutable codes and difference families

A cyclically permutable code is a binary code whose codewords are cyclically distinct and have full cyclic order. An important class of these codes are the constant weight cyclically permutable codes. In a code of this class all codewords have the same weight w. These codes have many applications, including in optical code-division multiple-access communication systems and in constructing protocol-sequence sets for the Mactive-out-of-T users collision channel without feedback. In this paper we construct optimal constant weight cyclically permutable codes with length 12, weight w, and a minimum Hamming distance 2w 2. Some of these codes coincide with the wellknown design called a difference family. Some of the constructions use combinatorial structures with other applications in coding.