Understanding Algebraic-Geometric Codes

Error-correcting codes derived from curves in an algebraic geometry are called Algebraic-Geometry Codes. The past couple of decades has seen extraordinary developments in the application of the ideas of algebraic geometry to the construction of codes and their decoding algorithms. This was initiated by the work of Goppa as generalizations of Bose-Chaudhuri-Hocquenghem (BCH), Reed-Solomon (RS), and Goppa codes. Goppa [10], [11] made the crucial observation by evaluating a set of rational functions at the points on an algebraic curve. In making this step, many of the tools needed to determine the important parameters of the code, or bounds on them, such as the code length, dimension, and minimum distance, already existed in the elegant theorem of algebraic geometry, notably the Hasse-Weil theorem and the Riemann-Roch theorem. The theory of Algebraic-geometry codes involves the relatively deep and fundamental results of algebraic geometry, and hence it requires effort on the part of the non expert to appreciate the significant developments of the area. The purpose of this report is not to survey the vast body of literature on algebraic-geometry codes, but to try and provide a short introduction to this subject. Hence, I will try to bypass most of the underlying algebraic geometry and provide only a brief overview of those concepts from algebraic geometry needed to appreciate the development. My hope to is write a report such that the other students in the class can understand it without too much trouble. Interested readers must refer to [1] for a much rigorous and complete survey of algebraic-geometry codes. However, there are several books that attempt to give self-contained treatments of algebraic geometry and codes ( [2], [5], [6]) and one should refer to them for proofs of various facts that are stated in this report. In the next section we briefly summarize some basic material regarding linear block codes on a general finite field Fq, of q elements. Following which we will review the construction of a basic class of codes, namely RS codes, in such a manner