  A coding scheme for the discrete memoryless broadcast channel with
{noiseless, noisy, generalized} feedback is proposed, and the associated
achievable region derived. The scheme is based on a block-Markov strategy
combining the Marton scheme and a lossy version of the Gray-Wyner scheme with
side-information. In each block the transmitter sends fresh data and update
information that allows the receivers to improve the channel outputs observed
in the previous block. For a generalization of Dueck's broadcast channel our
scheme achieves the noiseless-feedback capacity, which is strictly larger than
the no-feedback capacity. For a generalization of Blackwell's channel and when
the feedback is noiseless our new scheme achieves rate points that are outside
the no-feedback capacity region. It follows by a simple continuity argument
that for both these channels and when the feedback noise is sufficiently low,
our scheme improves on the no-feedback capacity even when the feedback is
noisy.
