  We describe a special instance of the Goerss-Hopkins obstruction theory, due
to Senger, for calculating the moduli of $E_\infty$ ring spectra with given
mod-$p$ homology. In particular, for the $2$-primary Brown-Peterson spectrum we
give a chain complex that calculates the first obstruction groups, locate the
first potential genuine obstructions, and discuss how some of the obstruction
classes can be interpreted in terms of secondary operations.
