UUnniivveerrssiittyy ooff PPeennnnssyyllvvaanniiaa SScchhoollaarrllyyCCoommmmoonnss Publicly Accessible Penn Dissertations 2013 AArriitthhmmeettiicc CCoonnssttrruuccttiioonnss OOff BBiinnaarryy SSeellff--DDuuaall CCooddeess Ying Zhang University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/edissertations Part of the Mathematics Commons RReeccoommmmeennddeedd CCiittaattiioonn Zhang, Ying, "Arithmetic Constructions Of Binary Self-Dual Codes" (2013). Publicly Accessible Penn Dissertations. 824. https://repository.upenn.edu/edissertations/824 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/edissertations/824 For more information, please contact [email protected]. AArriitthhmmeettiicc CCoonnssttrruuccttiioonnss OOff BBiinnaarryy SSeellff--DDuuaall CCooddeess AAbbssttrraacctt The goal of this thesis is to explore the interplay between binary self-dual codes and the \'etale cohomology of arithmetic schemes. Three constructions of binary self-dual codes with arithmetic origins are proposed and compared: Construction $\Q$, Construction G and the Equivariant Construction. In this thesis, we prove that up to equivalence, all binary self-dual codes of length at least $4$ can be obtained in Construction $\Q$. This inspires a purely combinatorial, non-recursive construction of binary self-dual codes, about which some interesting statistical questions are asked. Concrete examples of each of the three constructions are provided. The search for binary self-dual codes also leads to inspections of the cohomology ``ring" structure of the \'etale sheaf $\mu_2$ on an arithmetic scheme where $2$ is invertible. We study this ring structure of an elliptic curve over a $p$-adic local field, using a technique that is developed in the Equivariant Construction. DDeeggrreeee TTyyppee Dissertation DDeeggrreeee NNaammee Doctor of Philosophy (PhD) GGrraadduuaattee GGrroouupp Mathematics FFiirrsstt AAddvviissoorr Ted Chinburg KKeeyywwoorrddss Binary self-dual code, Etale cohomology SSuubbjjeecctt CCaatteeggoorriieess Mathematics This dissertation is available at ScholarlyCommons: https://repository.upenn.edu/edissertations/824 ARITHMETIC CONSTRUCTIONS OF BINARY SELF-DUAL CODES Ying Zhang A DISSERTATION in Mathematics Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy 2013 Supervisor of Dissertation Ted Chinburg, Professor of Mathematics Graduate Group Chairperson David Harbater, Professor of Mathematics Dissertation Committee Florian Pop, Professor of Mathematics Ted Chinburg, Professor of Mathematics Henry Towsner, Professor of Mathematics ARITHMETIC CONSTRUCTIONS OF BINARY SELF-DUAL CODES (cid:13)c COPYRIGHT 2013 Ying Zhang This work is licensed under the Creative Commons Attribution NonCommercial-ShareAlike 3.0 License To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/ Dedicated to my parents, Yongzhao Zhang and Shuyan Zhang iii Acknowledgments I would like to express my sincere thanks to my supervisor Ted Chinburg for his constant support and guidance throughout my graduate studies. His expertise and advice are invaluable to me in many ways beyond I could imagine. Among many other things, I have learned my first class in algebraic number theory from him, prepared for my qualify exam under his directions, and have had his motivating collaboration in my first publication. It has been a great pleasure to work with him. Without his helpful suggestions and instructions, this work would not have been possible. I would also like to thank Florian Pop and David Harbater for their constant help and illuminating discussions in mathematics. Henry Towsner, for serving on my thesis committee. David Zywina, Jonathan Block and Tony Pantev from whom I have benefited a lot from their classes. Robin Pemantle and Dennis DeTurck for giving me many helpful suggestions. Janet Burns, Monica Pallanti, Paula Scarbor- ough for their constant help in my daily life at the Math department, it is their dedication that makes the department such an enjoyable place to work at. iv I want to express my gratitude to Wenxuan Lu, whose passion for mathematics has always been encouraging to me. I have enjoyed my life as a graduate student at Penn to live in a big family of many helpful fellow graduate students: Zhentao Lu, Adam Topaz, Ryan Eberhart, Taisong Jing, Shanshan Ding, Hilaf Hasson, Justin Curry, Ryan Manion, Elaine So, Yang Liu, Ying Zhao, Zhaoting Wei, Tong Li, Haomin Wen, Xiaoxian Liu, Yiqing Cai and many others. I can not forget the many mornings, afternoons and evenings we spent in DRL discussing problems. I am also grateful for my friends in Philadelphia: Hao Sun, Yubin Bai, Zhaoxia Qian, Na Zhang, Rong Shen, Lai Jiang, Yang Jiao, Yin Xia, Chi Liu, Yan Wang, Naibo Chen, Yuyuan Liu, Shaohui Wang, Zuolu Liu. I am grateful for the many happy moments they brought in my life. Last but not least, I want to thank my parents Yongzhao Zhang and Shuyan Zhang for giving birth to me, raising me up and their unfailing love, understanding and support for me over the years. This thesis is dedicated to them. v ABSTRACT ARITHMETIC CONSTRUCTIONS OF BINARY SELF-DUAL CODES Ying Zhang Ted Chinburg The goal of this thesis is to explore the interplay between binary self-dual codes and the ´etale cohomology of arithmetic schemes. Three constructions of binary self-dual codes with arithmetic origins are proposed and compared: Construction Q, Construction G and the Equivariant Construction. In this thesis, we prove that up to equivalence, all binary self-dual codes of length at least 4 can be obtained in Construction Q. This inspires a purely combinatorial, non-recursive construction of binary self-dual codes, about which some interesting statistical questions are asked. Concrete examples of each of the three constructions are provided. The search for binary self-dual codes also leads to inspections of the cohomology “ring” structure of the ´etale sheaf µ on an arithmetic scheme where 2 is invertible. We study this 2 ring structure of an elliptic curve over a p-adic local field, using a technique that is developed in the Equivariant Construction. vi Contents 1 Introduction 1 2 Coding Theory Background 4 2.1 General Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Binary Self-dual Codes . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Extremal Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3 Construction Q 13 3.1 S-Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 Hilbert Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3 The Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.4 A Random Generation Algorithm . . . . . . . . . . . . . . . . . . . 23 4 Construction G 28 4.1 Arithmetic Duality of Global Fields . . . . . . . . . . . . . . . . . . 29 4.2 Duality of Arithmetic Schemes . . . . . . . . . . . . . . . . . . . . . 37 vii 5 The Equivariant Construction 41 5.1 Cohomology of a G-Complex . . . . . . . . . . . . . . . . . . . . . . 42 5.2 Equivariant Etale Sheaves . . . . . . . . . . . . . . . . . . . . . . . 45 5.3 The Localization Theorem . . . . . . . . . . . . . . . . . . . . . . . 48 5.4 The Equivariant Construction . . . . . . . . . . . . . . . . . . . . . 53 5.5 Comparison with Construction G . . . . . . . . . . . . . . . . . . . 58 5.6 The Smith Type Inequality . . . . . . . . . . . . . . . . . . . . . . 67 A Topological constructions of binary self-dual codes 75 viii