On the sharpness of the bound for the local converse theorem of p-adic GL_N, general N

Let F be a non-archimedean local field of characteristic zero. In this paper we construct examples of supercuspidal representations showing that the bound $[N\mathrm{/}2]$ for the local converse theorem of $G{L}_N(F)$ is sharp, N general, when the residual characteristic of $F$ is bigger than $N$.