A note on the Assmus--Mattson theorem for some ternary codes (a resume)

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