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Title:  Invariant embeddings of the DeligneLusztig curves with applications 
Author(s):  Eid, Abdulla 
Director of Research:  Duursma, Iwan M. 
Doctoral Committee Chair(s):  Nevins, Thomas A. 
Doctoral Committee Member(s):  Duursma, Iwan M.; Ullom, Stephen V.; Milkenovic, Olgica 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  DeligneLusztig Curves
Ree Curve Smooth Embeddings Nongaps semigroup Polar Codes Hash Families Tower of Function Fields 
Abstract:  In this thesis we study DeligneLusztig curves associated to the simple groups $\AAA$, $\BB$, and $\GG$ with applications in algebraic geometry codes, hash families, and polar codes. This thesis consists of four parts, the first part presents our result in finding smooth embeddings for the DeligneLusztig curves in projective space. The second part concerns certain RiemannRoch spaces and AGcodes from the Suzuki curve. The third part is about the construction of certain perfect hash families from towers of function fields. The fourth part is the study of the performance of binary concatenated algebraic geometry codes as polar codes. Let $C$ be a DeligneLusztig curve associated to a simple group $G$, i.e., $C$ is either the Hermitian, Suzuki, or Ree curve associated to the group $\AAA$, $\BB$, or $\GG$, respectively. In the first part of this thesis we apply the techniques that have been used for the Hermitian and Suzuki curves to find a very ample linear series for the Ree curve and to construct smooth embeddings for the three DeligneLusztig curves above in the projective space of dimension 2, 4, and 13, respectively. We provide a complete set of 5 equations that define the Suzuki curve in $\mathbb{P}^{4}$ and 105 equations that define the Ree curve in $\mathbb{P}^{13}$. These equations are then used to compute the Weierstrass nongaps semigroup for the Ree curve at $P_\infty$ over $\fg{27}$. In the second part we find an explicit basis for the RiemannRoch spaces $\LL(\ell (q^2+1)P_\infty)$ $(\ell \leq q^21)$ arising from the Suzuki curve. Then, we use these spaces to construct onepoint AGcodes with good parameters. In the third part we construct perfect and $\epsilon$almost strongly universal hash families from the recently constructed tower of function fields by Garcia, Stichtenoth, Bassa, and Beelen defined recursively by the equation $\Tri_{j}\left ( \sfrac{Y}{X^{q^k}}\right) + \Tri_{k}\left ( \sfrac{Y^{q^j}}{X}\right)=1$ over $\fg{q^n}$ ($n=j+k$). In the last part we study the performance of algebraic geometry codes as suitable kernels for channel polarization. We show that for a family of AGcodes of block length $L$ and kernel matrix $G_L$, that the exponent $E(G_L)\to 1$ as $L \to \infty$. We also compare how the binary concatenated ReedSolomon, Hermitian, and Suzuki codes behave as polar codes and as error correcting codes. In the these two settings, more geometry is preferable as the block size increases. 
Issue Date:  20130524 
URI:  http://hdl.handle.net/2142/44259 
Rights Information:  Copyright 2013 Abdulla Eid 
Date Available in IDEALS:  20130524 
Date Deposited:  201305 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois