  We address the problem of Reliable Broadcast in asynchronous message-passing
systems with $n$ nodes, of which up to $t$ are malicious (faulty), in addition
to a message adversary that can drop some of the messages sent by correct
(non-faulty) nodes.
  We present a Message-Adversary-Tolerant Byzantine Reliable Broadcast (MBRB)
algorithm that communicates an almost optimal amount of $O(|m|+n^2\kappa)$ bits
per node, where $|m|$ represents the length of the application message and
$\kappa=\Omega(\log n)$ is a security parameter. This improves upon the
state-of-the-art MBRB solution (Albouy, Frey, Raynal, and Ta\"iani, SSS 2021),
which incurs communication of $O(n|m|+n^2\kappa )$ bits per node.
  Our solution sends at most $4n^2$ messages overall, which is asymptotically
optimal. Reduced communication is achieved by employing coding techniques that
replace the need for all nodes to (re-)broadcast the entire message~$m$.
Instead, nodes forward authenticated fragments of the encoding of $m$ using an
erasure-correcting code. Under the cryptographic assumptions of PKI and
collision-resistant hash, and assuming $n > 3t + 2d$, where the adversary drops
at most~$d$ messages per broadcast, our algorithm allows most of the correct
nodes to reconstruct~$m$, despite missing fragments caused by the malicious
nodes and the message adversary.
