Essential dimension of simple algebras in positive characteristic

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Let p be a prime integer. For any integers 1 ⩽ s ⩽ r , ▪ p r , p s denotes the class of central simple algebras of degree p r and exponent dividing p s . For any s < r , we find a lower bound for the essential p-dimension of ▪ p r , p s . Furthermore, we compute an upper bound for ▪ 8 , 2 over a field of characteristic 2. As a result, we show ed 2 ( ▪ 4 , 2 ) = ed ( ▪ 4 , 2 ) = 3 and 3 ⩽ ed ( ▪ 8 , 2 ) ⩽ 10 over a field of characteristic 2.