  This paper focuses on the 1-to-K broadcast packet erasure channel (PEC),
which is a generalization of the broadcast binary erasure channel from the
binary symbol to that of arbitrary finite fields GF(q) with sufficiently large
q. We consider the setting in which the source node has instant feedback of the
channel outputs of the K receivers after each transmission. Such a setting
directly models network coded packet transmission in the downlink direction
with integrated feedback mechanisms (such as Automatic Repeat reQuest (ARQ)).
  The main results of this paper are: (i) The capacity region for general
1-to-3 broadcast PECs, and (ii) The capacity region for two classes of 1-to-K
broadcast PECs: the symmetric PECs, and the spatially independent PECs with
one-sided fairness constraints. This paper also develops (iii) A pair of outer
and inner bounds of the capacity region for arbitrary 1-to-K broadcast PECs,
which can be evaluated by any linear programming solver. For most practical
scenarios, the outer and inner bounds meet and thus jointly characterize the
capacity.
