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 Title: Invariant embeddings of the Deligne-Lusztig 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 Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Deligne-Lusztig Curves Ree Curve Smooth Embeddings Non-gaps semigroup Polar Codes Hash Families Tower of Function Fields Abstract: In this thesis we study Deligne-Lusztig 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 Deligne-Lusztig curves in projective space. The second part concerns certain Riemann-Roch spaces and AG-codes 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 Deligne-Lusztig 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 Deligne-Lusztig 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 non-gaps semigroup for the Ree curve at $P_\infty$ over $\fg{27}$. In the second part we find an explicit basis for the Riemann-Roch spaces $\LL(\ell (q^2+1)P_\infty)$ $(\ell \leq q^2-1)$ arising from the Suzuki curve. Then, we use these spaces to construct one-point AG-codes 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 AG-codes 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 Reed-Solomon, 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: 2013-05-24 URI: http://hdl.handle.net/2142/44259 Rights Information: Copyright 2013 Abdulla Eid Date Available in IDEALS: 2013-05-24 Date Deposited: 2013-05
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