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

Let $C$ be a four-weight binary code, which has all one vector. Furthermore, we assume that $C$ supports $t$-designs for all weights obtained from {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.