Optimal conflict-avoiding codes for three active users

We consider the problem to construct a code of the maximum cardinality which consists of binary vectors of length n with three ones and has the following property: a matrix of size 3 times n from any cyclic shifts of any three different code vectors contains the identity matrix of size 3 times 3 (with accuracy up to a permutation of columns). This property (in more general form) was considered in the connection with the problem to avoid conflicts in the channels of multiple access under a restriction to the number of active users (see L. A. Bassalygo and M. S. Pinsker, Problemy Peredachi Informatsii, 1983; J. L. Massey and P. Mathys, IEEE Trans. Inform. Theory , 1985; B.S. Tsybakov and A.R. Rubinov, Problems of Information Transmission , 2002). The cardinality of such a code corresponds to the number of all users, and this property means that each from any three active users can successfully transmit a packet of information in one of three attempts to do it during n slots of time without a collision with other active users. In particular, cyclic Steiner triple systems give examples of such conflict-avoiding codes if we choose representatives of the cyclic classes as code vectors. In the paper we present some constructions of conflict-avoiding codes of triples which are better as compared with those obtained from the cyclic Steiner triple systems