Error correcting codes with bayes decoder and optimized codebook

A framework for error correction coding that takes into account the difference in bit significance in the source symbols by using an appropriate error metric and minimizing it using a Bayes decoder and an optimized codebook. The Bayes decoder performs better than standard soft and hard minimum distance decoding and the optimized codebook performs better than classical linear block codes, e.g., Hamming codes. The error metric is a norm in information symbol space and is based on a loss function appropriately defined according to an approach for assigning significance to the various bits in the source bit stream. The Bayes decoder of this metric is defined and an optimized codebook generated that optimizes this metric under a noisy channel. The framework for error correction coding is implemented for increased reduncancy in a communications system or a data storage system and is optimized to combat noise in such systems.