We present a robust and high-payload black-box multi-bit watermarking scheme for generative models. In order to embed a high payload message while retaining robustness against modifications of the watermarked network, we rely on the use of channel codes with strong error correction capacity (polar codes). This, in turn, increases the number of (coded) bits to be embedded within the network, thus challenging the embedding capabilities of the watermarking scheme. For this reason, we split the watermark bits into several chunks, each of which is associated with a different watermark triggering input. Through extensive experiments on the StyleGAN family of generative models, we show that the proposed method has excellent payload and robustness performance, allowing great flexibility to trade off between payload and robustness. Noticeably, our method demonstrates the capability of embedding over 100,000 coded bits for a net payload of up to 8192 bits while maintaining high image quality, with a PSNR exceeding 37 dB. Experiments demonstrate that the proposed high-payload strategy effectively improves the robustness of messages via high-performance channel codes, against white-box model attacks such as fine-tuning and pruning. Codes at: https://github.com/jumpycat/CCMark