Math and Computer Science Honors Theses

Toric and Quasi-Toric Codes

Date of Creation

5-26-2005

Document Type

Departmental Honors Thesis - Restricted Access

Department

Mathematics

First Advisor

John B. Little

Abstract

Coding theory, in its most general sense, is the study of methods for the efficient and accurate transfer of information from one place to another through a communication channel. These channels, however, are never perfect. Errors in transmission do occur and must routinely be dealt with. In fact, the most effective way to reduce the number of errors in a received message is to utilize an error-correcting code. Consequently, NASA has adopted certain error-correcting codes for deep space communication just as record labels have turned to others for the purpose of CD recording. These are only a few of the many applications of coding theory. We primarily concentrate on two very closely related error-correcting code constructions: toric and "quasi"-toric codes. Using concepts from abstract algebra, we first construct a finite field using an irreducible polynomial. We then choose configurations of monomials over the field. The evaluation of each monomial at every point in the field yields the codewords corresponding to each of the monomials. By using elements from linear algebra, we find and explain the error-correction capability of selected toric codes. Using Pick's Theorem and a theorem relating affine and monomial equivalence, we classify all 2-D toric codes with up to six monomial evaluation points. We also establish the notion of families of toric codes and present formulas for determining their error-correction capability. Thus, for any square, isosceles triangular, and nonisosceles triangular collection of points in a finite field with perpendicular legs, we can determine the error-correction capability of the corresponding toric code. In addition, we discover that adjusting the toric code definition gives us the similar "quasi"-toric construction, with greatly improved error-correction capability. As a result, these "quasi"-toric codes are consistently as good as or better than the best previously known existing codes in terms of error-correction.

Comments

Reader: Sharon Frechette

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