Note on the tight relative 2-designs on H(n,2)

Bannai-Bannai-Bannai studied intensively the tight relative 2-designs on two shells in H(n,2) and gained many important results. Their results indicate the relationship among relative t-designs in binary Hamming association schemes, combinatorial t-designs and Hadamard matrices. In this note we study the further relationship between tight relative 2-designs on two shells in H(n,2) and them. We first prove that the existence of a special tight relative 2-design of H(n,2) on two shells is equivalent to that of a Hadamard matrix of size m+1, where m≡−1(mod4) and n=2m; we also show how tight relative 2-designs of H(n,2) on two shells and BIBD or λ-linked designs are related. Next, for each integer n, where 30