  A relative t-design in the binary Hamming association schemes H(n,2) is
equivalent to a weighted regular t-wise balanced design, i.e., certain
combinatorial t-design which allow different sizes of blocks and a weight
function on blocks. In this paper, we study relative t-designs in H(n,2),
putting emphasis on Fisher type inequalities and the existence of tight
relative t-designs. We mostly consider relative t-designs on two shells. We
prove that if the weight function is constant on each shell of a relative
t-design on two shells then the subset in each shell must be a combinatorial
(t-1)-design. This is a generalization of the result of Kageyama who proved
this under the stronger assumption that the weight function is constant on the
whole block set. Using this, we define tight relative t-designs for odd t, and
a strong restriction on the possible parameters of tight relative t-designs in
H(n,2). We obtained a new family of such tight relative t-designs, which were
unnoticed before. We will give a list of feasible parameters of such relative
3-designs with n up to 100, and then we discuss the existence and/or the
non-existence of such tight relative 3-designs. We also discuss feasible
parameters of tight relative 4-designs on two shells in H(n,2) with n up 50. In
this study we come up with the connection on the topics of classical design
theory, such as symmetric 2-designs (in particular 2-(4u-1,2u-1,u-1) Hadamard
designs) and Driessen's result on the non-existence of certain 3-designs. We
believe the Problem 1 and Problem 2 presented in Section 5.2 open a new way to
study relative t-designs in H(n,2). We conclude our paper listing several open
problems.
