Essential dimension of simple algebras in positive characteristic

Let $p$ be a prime integer, $1\le s\le r$ integers, $F$ a field of characteristic $p$. Let $\text{cat}{Dec}_{p^r}$ denote the class of the tensor product of $r$ $p$-symbols and $\text{cat}{Alg}_{p^r,p^s}$ denote the class of central simple algebras of degree $p^r$ and exponent dividing $p^s$. For any integers $s<r$, we find a lower bound for the essential $p$-dimension of $\text{cat}{Alg}_{p^r,p^s}$. Furthermore, we compute upper bounds for $\text{cat}{Dec}_{p^r}$ and $\text{cat}{Alg}_{8,2}$ over $\mathrm{ch}(F)=p$ and $\mathrm{ch}(F)=2$, respectively. As a result, we show ${\text{ed}}_2(\text{cat}{Alg}_{4,2})=\text{ed}(\text{cat}{Alg}_{4,2})={\text{ed}}_2({\text{gGL}}_4\mathrm{/}{\text{gmu}}_2)=\text{ed}({\text{gGL}}_4\mathrm{/}{\text{gmu}}_2)=3$ and $3\le \text{ed}(\text{cat}{Alg}_{8,2})=\text{ed}({\text{gGL}}_8\mathrm{/}{\text{gmu}}_2)\le 10$ over a field of characteristic 2.