Large prime-power-field representations of n-color algebras

Determine whether an $n$-color algebra has a representation over a finite field of order $p^k$, with $k>1$ and $p^k>n^4+5$; if no such representation exists, derive an obstruction distinct from the prime-field sum-product argument.

Background

The paper’s earlier finite-field analysis establishes an $n^4+5$ upper bound for representations over prime fields, using the sum-product phenomenon in fields of prime order. The author explicitly notes that this argument does not extend to prime-power fields with nontrivial extensions, since those fields do not exhibit the same phenomenon in the required form.

The unresolved issue is whether representations over fields of order $p^k$ can exist beyond the prime-field bound, or whether a different argument can rule them out.