Error correcting codes make possible much of digital technology, from modems to cell phones to compact disk players. The problem is to send a message across a noisy channel with perfect fidelity, or at least with as much fidelity as possible. Below we will describe the theory of these codes. It is based on some ideas from “modern algebra,” including finite number systems called rings and fields...

This thesis deals with quantum error correcting codes. In first two chapters necessary introduction to quantum computation and classical error correction is presented. Previous results on construction of quantum error correcting codes are presented in the third and fourth chapter. Mainly Calderbank-Steane-Shor (CSS) codes and stabilizer codes are discussed together with the introduction to codi...

These notes are a record of proceedings in the QMW Combinat-orics Study Group in November and December 1998. Since we are discrete mathematicians and know little quantum theory, the notes are not strong on the physics background (but we give references to several sources for this). We have tried to compare quantum with classical error correction where possible, and to provide enough information...

Markus GRASSL received his diploma degree in Computer Science in 1994 and his doctoral degree in 2001, both from the Fakultät für Informatik, Universität Karlsruhe (TH), Germany. From 1994 to 2007 he was a member of the Institut für Algorithmen und Kognitive Systeme, Fakultät für Informatik, Universität Karlsruhe (TH), Germany. From 2007 to 2008 he was with the Institute for Quantum Optics and ...

is an encoder, that is, regarding Fq as the set of words in a natural language , the function V tells how to encode the words. The matrix V = (vij) that describes the linear map V is called a generating matrix of the code. We say that V is in standard form if V = (IdP ) for a matrix P , that is vij = δij , for i, j = 1, . . . , d (where δij = 0, i 6= j, δii = 1). In this case we say that the fi...

Automatic text categorization has become a vital topic in many applications. Imagine for example the automatic classi cation of Internet pages for a search engine database. The traditional 1-of-n output coding for classi cation scheme needs resources increasing linearly with the number of classes. A di erent solution uses an error correcting code, increasing in length with O(log2(n)) only. In t...

Automatic text categorization has become a vital topic in many applications. Imagine for example the automatic classification of Internet pages for a search engine database. The traditional 1-of-n output coding for classification scheme needs resources increasing linearly with the number of classes. A different solution uses an error correcting code, increasing in length with O(log2(n)) only. I...

We investigate one-point algebraic geometric codes CL(D, G) associated to maximal curves recently characterized by Tafazolian and Torres given by the affine equation yl = f(x), where f(x) is a separable polynomial of degree r relatively prime to l. We mainly focus on the curve y4 = x3 +x and Picard curves given by the equations y3 = x4-x and y3 = x4 -1. As a result, we obtain exact value of min...