Many broadcast algorithms have been proposed for the mesh over the past decade. However, most of these algorithms do not exhibit good scalability properties as the network size increases. As a consequence, most existing broadcast algorithms cannot support real-world parallel applications that require large-scale system sizes due to their high computational demands. Motivated by these observations, this study proposes a new adaptive broadcast algorithm for the mesh. The unique feature of our algorithm is that it handles broadcast operations with a fixed number of message-passing steps irrespective of the network size. Our algorithm is based on the coded path routing, which has been proposed in [1]. Results from extensive comparative analysis reveal that the proposed algorithm exhibits superior performance characteristics over those of the well-known Recursive Doubling and Extending Dominating Node algorithms.