A note on the Assmus--Mattson theorem for some binary codes II

Let $C$ be a four-weight code, which has all one vector. Furthermore, we assume that $C_\ell$ are $t$-designs for all $\ell$ by the Assmus--Mattson theorem. We previously showed that $t\leq 5$. In the present paper, we show an analogue of this result in the cases of five and six-weight codes.