Collaborative perception empowers each agent to improve its perceptual ability through the exchange of perceptual messages with other agents. It inherently results in a fundamental trade-off between perception ability and communication cost. To address this bottleneck issue, our core idea is to optimize the collaborative messages from two key aspects: representation and selection. The proposed codebook-based message representation enables the transmission of integer codes, rather than high-dimensional feature maps. The proposed information-filling-driven message selection optimizes local messages to collectively fill each agent's information demand, preventing information overflow among multiple agents. By integrating these two designs, we propose CodeFilling, a novel communication-efficient collaborative perception system, which significantly advances the perception-communication trade-off and is inclusive to both homogeneous and heterogeneous collaboration settings. We evaluate CodeFilling in both a real-world dataset, DAIR-V2X, and a new simulation dataset, OPV2VH+. Results show that CodeFilling outperforms previous SOTA Where2comm on DAIR-V2X/OPV2VH+ with 1,333/1,206 times lower communication volume. Our code is available at~\url{https://github.com/PhyllisH/CodeFilling}.