  We consider the free boundary condition Gibbs measure of the Potts model on a
random tree. We provide an explicit temperature interval below the
ferromagnetic transition temperature for which this measure is extremal,
improving older bounds of Mossel and Peres. In information theoretic language
extremality of the Gibbs measure corresponds to non-reconstructability for
symmetric q-ary channels. The bounds are optimal for the Ising model and appear
to be close to what we conjecture to be the true values up to a factor of
0.0150 in the case q = 3 and 0.0365 for q = 4. Our proof uses an iteration of
random boundary entropies from the outside of the tree to the inside, along
with a symmetrization argument.
