In Shioda's famous paper "[An Example of Unirational Surfaces in Characteristic $p$](https://link.springer.com/article/10.1007%2FBF01350715)" ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=374149)), the author proved that the Fermat surface over the characteristic-$p$ field $k$
$$
x\_1^n +x\_2^n + x\_3^n+x\_4^n =0
$$
is unirational if $q \equiv -1 \pmod n$ for some $p$-power $q$.

At end of this paper, he mentioned that he proved it in the higher-dimension case with similar method, *i.e.*,
$$ x\_1^n +x\_2^n + \cdots +x\_{m+1}^n =0$$
is unirational if $m \geq 3$ is odd and $q \equiv -1 \pmod n$ for some $p$-power $q$. But I can't prove it with his method in the surface case. How does his method generalize?