Multiple-message broadcast, or k-broadcast, is one of the fundamental problems in network communication. In short, there are k packets distributed across the network, each of them has to be delivered to all other nodes. We consider this task in the model of multi-hop radio network, in which n nodes interact by transmitting and receiving messages. A message transmitted at a round reaches all neighbors of the transmitter at the end of the same round, but may not be successfully received by some, or even all, of these neighbors. More specifically, a node receives a message at a round if this is the only message that has reached this node in this round. Due to this specific interference-prone nature of radio networks, many communication tasks become more challenging and more costly than in other types of networks, especially in ad-hoc setting in which each node knows only its own id and linear estimates on the basic network parameters, such as the number of nodes n, diameter D and maximum node degree Δ. We design a new randomized k-broadcast algorithm combining the bestof two worlds: efficient randomized transmission schedules and network coding. We show that our algorithm accomplishes multi-broadcast in O(log Δ) amortized number of communication rounds per packet, with high probability. This improves over the best previous solution of Bar-Yehuda, Israeli and Itai, which guarantees only O(log Δ log n) of amortized number of rounds per packet, with high probability.