  Let $p$ be a prime integer, $1\leq s\leq r$ integers, $F$ a field of
characteristic $p$. Let $\cat{Dec}_{p^r}$ denote the class of the tensor
product of $r$ $p$-symbols and $\cat{Alg}_{p^r,p^s}$ 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
$\cat{Alg}_{p^r,p^s}$. Furthermore, we compute upper bounds for
$\cat{Dec}_{p^r}$ and $\cat{Alg}_{8,2}$ over $\ch(F)=p$ and $\ch(F)=2$,
respectively. As a result, we show
$\ed_{2}(\cat{Alg}_{4,2})=\ed(\cat{Alg}_{4,2})=\ed_{2}(\gGL_{4}/\gmu_{2})=\ed(\gGL_{4}/\gmu_{2})=3$
and $3\leq \ed(\cat{Alg}_{8,2})=\ed(\gGL_{8}/\gmu_{2})\leq 10$ over a field of
characteristic 2.
