A note on the Assmus–Mattson theorem for some ternary codes

Let C be a two and three-weight ternary code. Furthermore, we assume that \(C_\ell \) $C_{\ell }$ are t-designs for all \(\ell \) $\ell$ by the Assmus–Mattson theorem. We show that \(t \le 5\) $t\le 5$ . As a corollary, we provide a new characterization of the (extended) ternary Golay code.