This paper describes a robust data embedding scheme which uses noise resilient channel codes based on multidimensional lattice structures. Compared to prior work in digital watermarking, the proposed scheme can handle a significantly larger quantity of signature data such as gray-scale or color images. A trade-off between the quantity of hidden data and the quality of the watermarked image is achieved by varying the number of quantization levels for the signature, and a scale factor for data embedding. Experimental results on signature recovery from JPEG compressed watermarked images show that good quality reconstruction is possible even when the images are lossy compressed by as much as 85%. Potential applications of this method include, in addition to watermarking, digital data hiding for security and for bit stream control and manipulation.