Algebraic-geometric Codes and Modernized Algebraic Decoding

Algebraic-geometric (AG) code has long been identified as a possible candidate to replace the ReedSolomon (RS) code for error correction in both wireless communications and storage devices. In this presentation, the construction of an AG code will be introduced with a discussion of its application advantages. An algebraic soft decoding (ASD) algorithm is designed for the AG code. With an error correction capability beyond the half distance boundary, the ASD algorithm is one of the most powerful decoding algorithms with a moderate decoding complexity. For implementation, a number of complexity reduction approaches will also be introduced. Performance on ASD decoding of AG code will be shown, comparing with RS codes defined in different sizes of finite field.

Aiming to engineer the algorithm, this presentation will also introduce a modernized ASD algorithm, named the progressive ASD (PASD) algorithm. It is shown instead of performing the complexity dominated interpolation process with a fixed decoding parameter, the ASD algorithm can be performed in an iterative way, enhancing its decoding capability progressively. Like most of the modern decoding approaches, for example, the belief propagation (BP) algorithm, the PASD algorithm can smartly adjust both of its decoding capability and complexity according to the quality of the received information. Surprisingly, more than reducing the decoding complexity, the PASD algorithm can also slightly outperform the existing ASD algorithm. This thanks to its ability of sensitively identifying the successful decoding event.