Valuations on tensor powers of a division algebra

We study the following question in this paper: If p is a prime, m a positive integer, and S=(sm,…,s1) an arbitrary sequence consisting of “Y” or “N,” does there exist a division algebra of exponent pm over a valued field (F,v) such that the underlying division algebra of the tensor power D⊗pi has a valuation extending v if and only if sm−i=Y? We show that if such an algebra exists, then its index must be bounded below by a power of p that depends on both m and S, and we then answer the question affirmatively by constructing such an algebra of minimal index.