Coding Theory with Methods from Algebraic Geometry and Number Theory

University Of Nebraska-Lincoln, Lincoln NE

Investigators

Abstract

In previous work, the PI introduced the study of algebraic geometric codes over local Artinian rings, and the first main goal of this project is to further the understanding of these objects. In particular, the investigator studies the minimum squared Euclidean weight of these codes. This quantity is closely related to an exponential sum, where the sum is over points on the curve (defined over the ring) used in the construction of the code. Thus, the question of the error-correcting capability of these codes is reduced to a question in number theory. Results have already been obtained by the investigator and a colleague in the two special cases where the curve involved is either the Serre-Tate canonical lift of an ordinary elliptic curve or a plane curve with a single point at infinity. The PI is continuing this work with the goal of studying the squared Euclidean weight and associated exponential sum for more general curves. Additionally, the PI is working towards the development of a decoding algorithm for these codes, with respect to the squared Euclidean weight. The second main goal of the project involves the search for a structure theory of codes, a topic which has been around almost from the beginning of the study of codes. This search was largely unsuccessful until the introduction of so-called critical indecomposable codes in a recent paper of Assmus. The PI is developing this theory further and demonstrating its power by revisiting the classification of self-dual codes. Whenever data is transmitted across a channel, errors are bound to occur. The goal of coding theory is to find efficient ways of adding redundancy so that errors can be detected, or even corrected. Two basic questions must be asked about every code: "What is the error-correcting capability of this code?" and "If an error occurs in transmission while using this code, is there an efficient way of recovering the original codeword?" In the case that code is defined over the ring of integers modulo a power of a prime, the error-correcting capability of the code is measured in terms of the code's minimum squared Euclidean weight and the investigator studies this property in the case of algebraic geometric codes. The problem of recovering the original codeword is equivalent to finding a decoding algorithm for the code, which has important connections to cryptography also. Finally, a structure theory for codes allows the study of codes from a systematic point of view. In particular, self-dual codes often achieve a good balance between error-correcting capability and efficiency, and a structure theory allows a new attack for the study of these objects.