Upper bound of multiplicity in Cohen-Macaulay rings of prime characteristic

Let $(R,\mathfrak{m})$ be a local ring of prime characteristic $p$ and of dimension $d$ with the embedding dimension $v$, type $s$ and the Frobenius test exponent for parameter ideals $\mathrm{F}\mathrm{t}\mathrm{e}(R)$. We will give an upper bound for the multiplicity of Cohen-Macaulay rings in prime characteristic in terms of $\mathrm{F}\mathrm{t}\mathrm{e}(R),d,v$ and $s$. Our result extends the main results for Gorenstein rings due to Huneke and Watanabe.