${\mathbb{Z}}_p$-torus actions on positively curved manifolds

In this article, we study closed, positively curved $n$-manifolds that admit an effective, isometric ${\mathbb{Z}}_p^r$-action with a fixed point, where $p$ is an odd prime. For all sufficiently large $n$, we obtain a symmetry-rank bound in Theorem A that improves the $3n\mathrm{/}8$ bound of Fang and Rong and of Ghazawneh. We improve on this bound for small odd primes $3\le p\le 19$ in Theorem B. One of our main tools comes from the theory of error-correcting codes and is of independent interest: we derive a finite-length Plotkin bound and a finite-length Elias-Bassalygo bound for $q$-ary codes and show that the finite-length Plotkin bound is asymptotically sharper for all primes $p\ge 23$.