{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/44259"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/44259","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Invariant embeddings of the Deligne-Lusztig curves with applications","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.","abstract_html":"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 <span class=\"etd-inline-math\">\\mathbb{P}<sup>4</sup></span> and 105 equations that define the Ree curve in <span class=\"etd-inline-math\">\\mathbb{P}<sup>13</sup></span>. These equations are then used to compute the Weierstrass non-gaps semigroup for the Ree curve at <span class=\"etd-inline-math\">P<sub>\\</sub>infty</span> over $\\fg{27}$. In the second part we find an explicit basis for the Riemann-Roch spaces <span class=\"etd-inline-math\">\\LL(\\ell (q<sup>2</sup>+1)P<sub>\\</sub>infty)</span> <span class=\"etd-inline-math\">(\\ell \\leq q<sup>2</sup>-1)</span> 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 <span class=\"etd-inline-math\">&epsilon;</span>-almost strongly universal hash families from the recently constructed tower of function fields by Garcia, Stichtenoth, Bassa, and Beelen defined recursively by the equation <span class=\"etd-inline-math\">\\Tri<sub>j</sub>\\left ( \\sfrac{Y}{X<sup>q<sup>k</sup></sup>}\\right) + \\Tri<sub>k</sub>\\left ( \\sfrac{Y<sup>q<sup>j</sup></sup>}{X}\\right)=1</span> over <span class=\"etd-inline-math\">\\fg{q<sup>n</sup>}</span> ($n=j+k$). 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 <span class=\"etd-inline-math\">G<sub>L</sub></span>, that the exponent <span class=\"etd-inline-math\">E(G<sub>L</sub>)\\to 1</span> 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.","abstract_has_math":true,"creators":["Eid, Abdulla"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Duursma, Iwan M.","Nevins, Thomas A.","Ullom, Stephen V.","Milkenovic, Olgica"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2013,"date_issued":"2013-05-24T22:05:47Z","date_published":"2013-05-24T22:05:47Z","updated_at":"2026-07-22T22:25:34Z","subjects":["Deligne-Lusztig Curves","Ree Curve","Smooth Embeddings","Non-gaps semigroup","Polar Codes","Hash Families","Tower of Function Fields"],"languages":["en"],"rights":["Copyright 2013 Abdulla Eid"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/44259","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Duursma, Iwan M.","Nevins, Thomas A.","Ullom, Stephen V.","Milkenovic, Olgica"]},{"key":"dc:creator","label":"Author","values":["Eid, Abdulla"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2013-05-24T22:05:47Z","2013-05"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Deligne-Lusztig Curves","Ree Curve","Smooth Embeddings","Non-gaps semigroup","Polar Codes","Hash Families","Tower of Function Fields"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2013 Abdulla Eid"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/44259"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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.","Item withdrawn by Alexis Thompson (athmpsn1@illinois.edu) on 2013-04-16T21:14:04Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 Eid_Abdulla.pdf: 833156 bytes, checksum: 41c5396e40a3f2b61fadb269c345a0e0 (MD5)","Made available in DSpace on 2013-05-24T22:05:47Z (GMT). No. of bitstreams: 2 Abdulla_Eid.pdf: 833156 bytes, checksum: 41c5396e40a3f2b61fadb269c345a0e0 (MD5) license.txt: 4057 bytes, checksum: a7d68c726f0d7040424fbf1aa64f174c (MD5)"]},{"key":"dc:title","label":"Title","values":["Invariant embeddings of the Deligne-Lusztig curves with applications"]}]}],"canonical_facts":{"dc:contributor":["Duursma, Iwan M.","Nevins, Thomas A.","Ullom, Stephen V.","Milkenovic, Olgica"],"dc:creator":["Eid, Abdulla"],"dc:date":["2013-05-24T22:05:47Z","2013-05"],"dc:description":["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.","Item withdrawn by Alexis Thompson (athmpsn1@illinois.edu) on 2013-04-16T21:14:04Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 Eid_Abdulla.pdf: 833156 bytes, checksum: 41c5396e40a3f2b61fadb269c345a0e0 (MD5)","Made available in DSpace on 2013-05-24T22:05:47Z (GMT). No. of bitstreams: 2 Abdulla_Eid.pdf: 833156 bytes, checksum: 41c5396e40a3f2b61fadb269c345a0e0 (MD5) license.txt: 4057 bytes, checksum: a7d68c726f0d7040424fbf1aa64f174c (MD5)"],"dc:identifier":["http://hdl.handle.net/2142/44259"],"dc:language":["en"],"dc:rights":["Copyright 2013 Abdulla Eid"],"dc:subject":["Deligne-Lusztig Curves","Ree Curve","Smooth Embeddings","Non-gaps semigroup","Polar Codes","Hash Families","Tower of Function Fields"],"dc:title":["Invariant embeddings of the Deligne-Lusztig curves with applications"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:34Z"}