We address the problem of robust coding in which the signal information should be preserved in spite of intrinsic noise in the representation. We present a theoretical analysis for 1- and 2-D cases and characterize the optimal linear encoder and decoder in the mean-squared error sense. Our analysis allows for an arbitrary number of coding units, thus including both under- and over-complete representations, and provides insights into optimal coding strategies. In particular, we show how the form of the code adapts to the number of coding units and to different data and noise conditions in order to achieve robustness. We also present numerical solutions of robust coding for high-dimensional image data, demonstrating that these codes are substantially more robust than other linear image coding methods such as PCA, ICA, and wavelets.