On p-blocks of symmetric and alternating groups with all irreducible Brauer characters of prime power degree

We classify all p-blocks of the symmetric and alternating groups where all irreducible p-Brauer characters are of prime power degree as well as the p-blocks of these groups where all p-Brauer characters are of p-power degree. This is an extension and a block refinement of an earlier result where all irreducible p-Brauer characters were assumed to be of p-power degree.