  Let $L_{mkl}\subset \mathbb{R}^{m+k+l}$ be the set of vectors which have $m$
of entries $-1$, $k$ of entries $0$, and $l$ of entries $1$. In this paper, we
investigate the largest subset of $L_{mkl}$ whose diameter is smaller than that
of $L_{mkl}$. The largest subsets for $m=1$, $l=2$, and any $k$ will be
classified. From this result, we can classify the largest $4$-distance sets
containing the Euclidean representation of the Johnson scheme $J(9,4)$. This
was an open problem in Bannai, Sato, and Shigezumi (2012).
