Mutually orthogonal matrices from division algebras

Matrices A and B in $M_n(\mathbb{C})$ are said to be mutually orthogonal if $A{B}^{?}+B{A}^{?}=0$, where ^? denotes the conjugate transpose. We study cardinalities of certain $\mathbb{R}$-linearly independent families of matrices arising from matrix embeddings of a division algebra of index m with center a number field Z, satisfying the property that matrices from different families are mutually orthogonal. The question is of importance in the context of coding for certain wireless channels, where the cardinalities of such sets is connected to the maximum code rate consistent with low decoding complexity. It follows from our results that the maximum code rate for the codes we consider is severely limited.