Error control is essential in all digital communications or storage, ranging from bank transactions to internet commerce, from CD or DVD to cloud computing, from cellphone conversations to outer space explorations, among many other applications. Many practical error correcting codes are constructed via algebraic curves over finite fields. These codes have been extensively studied from both theoretical and practical point of views. However, many questions still remain open, particularly on code structures, code constructions and decoding complexity. This project will strive to study structural properties and decoding algorithms for codes from algebraic geometry with the aim of making these codes more amenable to applications. The project is multi-disciplinary lying at the crossroads of mathematics, computer science,and electronic engineering. It bridges pure mathematics, particularly discrete mathematics and algebraic geometry, with practical applications in digital communications. Any new result or any good algorithm for codes could be used to improve communication capability in practice.
Algebraic geometry (AG) codes have a tremendous amount of algebraic structure. Exploiting this algebraic structure enables construction of codes, efficient encoding, and efficient decoding. While many advances in decoding algorithms for AG codes has been made in the last decade, most of these apply only to one-point AG codes. Multipoint codes can have much better parameters than comparable one-point codes. This project will study how to realize this advantage in term of encoding and decoding algorithms. Another important issue relates to explicit constructions of bases for AG codes, especially those from curves in higher dimensional spaces (function field towers). This project studies both list decoding beyond half distance and fast unique decoding below half distance. Techniques include power series and Grobner bases. For many AG codes, even if decoding is only up to half distance, their error control capability is much higher than Reed-Solomon codes which are widely used in practice. To make AG codes more suitable for practical implementations, the project aims to reduce the decoding complexity and memory requirements via power series representations.
