In this paper, we consider the self-similar measure $ν_λ=\mathrm{law}\left(\sum_{j \geq 0} ξ_j λ^j\right)$ on $\mathbb{R}$, where $|λ|<1$ and the $ξ_j \sim ν$ are independent, identically distributed with respect to a measure $ν$ finitely supported on $\mathbb{Z}$. One example of this is the classical Bernoulli convolution. It is known that for certain combinations of algebraic $λ$ and $ν$ uniform on an interval, $ν_λ$ is absolutely continuous and its Fourier transform has power decay (\cite{garsia1}, \cite{feng}); in the proof, it is exploited that for these combinations, a quantity called the Garsia entropy $h_λ(ν)$ is maximal. We show that absolute continuity and power Fourier decay occur when $λ$ and $ν$ are such that $h_λ(ν)$ is maximal and classify all combinations for which this is the case. We find that if an algebraic $λ$ without a Galois conjugate of modulus exactly one has a $ν$ such that $h_λ(ν)$ is maximal, then all Galois conjugates of $λ$ must be smaller in modulus than one and $ν$ must satisfy a certain finite set of linear equations in terms of $λ$.