Tight Relative 2- and 4-Designs on Binary Hamming Association Schemes

The concept of relative t-designs for Q-polynomial association schemes is due to Delsarte (Philips Res Rep 32:373–411, 1997). We formulate Fisher type lower bounds as well as the concept of tight relative 2- and 4-designs very explicitly for binary Hamming association schemes H(n, 2). Then we discuss some examples of tight relative 2- and 4-designs on H(n, 2) in this sense, in connection with the work of Woodall (Proc Lond Math Soc (3) 20:669–687, 1970), Enomoto-Ito-Noda (Osaka J Math 16:39–43, 1979), and others. The present paper is expected to be the first of a series of our studies of tight relative t-designs on Q-polynomial association schemes in a more general context.