Additional properties of parity based bit-counting complexity classes and hierarchies

We study some properties of the parity based bit-counting complexity classes ${\bf B_{|0| \oplus}P}$ and ${\bf B_{|1| \oplus}P}$. We first prove that both of these complexity classes are closed under complement and ${\bf B_{|1|\oplus}P}\subseteq {\bf B_{|0|\oplus}P}$. We then prove that ${\bf US}\su