On the existence of tight relative 2-designs on binary Hamming association schemes

It is known that there is a close analogy between “Euclidean t -designs vs. spherical t -designs” and “Relative t -designs in binary Hamming association schemes vs. combinatorial t -designs”. In this paper, we want to prove how much we can develop a similar theory in the latter situation, imitating the theory in the former one. We first prove that the weight function is constant on each shell for tight relative t -designs on p shells on a wide class of Q -polynomial association schemes, including Hamming association schemes. In the theory of Euclidean t -designs on 2 concentric spheres (shells), it is known that the structure of coherent configurations is naturally attached. However, it seems difficult to prove this claim in a general context. In the case of tight 2-designs in combinatorial 2-designs, there are great many tight 2-designs, i.e., symmetric 2-designs, while there are very few tight 2 e -designs for e ≥ 2 . So, as a starting point, we concentrate our study to the existence problem of tight relative 2-designs, in particular on 2 shells, in binary Hamming association schemes H ( n , 2 ) . We prove that every tight relative 2-design on 2 shells in H ( n , 2 ) has the structure of coherent configuration. We determine all the possible parameters of coherent configurations attached to such tight relative 2-designs for n ≤ 30 . Moreover for each of them we determine whether there exists such a tight relative 2-design or not, either by constructing them from symmetric 2-designs or Hadamard matrices, or theoretically showing the non-existence. In particular, we show that for n ≡ 6 ( mod 8 ) , there exist such tight relative 2-designs whose weight functions are not constant. These are the first examples of those with non-constant weight.