  The unitary Birkhoff theorem states that any unitary matrix with all row sums
and all column sums equal unity can be decomposed as a weighted sum of
permutation matrices, such that both the sum of the weights and the sum of the
squared moduli of the weights are equal to unity. If the dimension~$n$ of the
unitary matrix equals a power of a prime $p$, i.e.\ if $n=p^w$, then the
Birkhoff decomposition does not need all $n!$ possible permutation matrices, as
the epicirculant permutation matrices suffice. This group of permutation
matrices is isomorphic to the general affine group GA($w,p$) of order only
$p^w(p^w-1)(p^w-p)...(p^w-p^{w-1}) \ll \left( p^w \right)!$.
