Citation
Schroeder, Brian Leroy (2010) On Elliptic Semiplanes, an Algebraic Problem in Matrix Theory, and Weight Enumeration of Certain Binary Cyclic Codes. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/84VV-S966. https://resolver.caltech.edu/CaltechTHESIS:10262009-141148765
Abstract
An elliptic semiplane is a λ-fold of a symmetric 2-(v,k,λ) design, where parallelism is transitive. We prove existence and uniqueness of a 3-fold cover of a 2-(15,7,3) design, and give several constructions. Then we prove that the automorphism group is 3.Alt(7). The corresponding bipartite graph is a minimal graph with valency 7 and girth 6, which has automorphism group 3.Sym(7).
A polynomial with real coefficients is called formally positive if all of the coefficients are positive. We conjecture that the determinant of a matrix appearing in the proof of the van der Waerden conjecture due to Egorychev is formally positive in all cases, and we prove a restricted version of this conjecture. This is closely related to a problem concerning a certain generalization of Latin rectangles.
Let ω be a primitive nth root of unity over GF(2), and let mi(x) be the minimal polynomial of ωi. The code of length n = 2r-1 generated by m1(x)mt(x) is denoted Crt. We give a recursive formula for the number of codewords of weight 4 in Cr11r for each r.
Item Type: | Thesis (Dissertation (Ph.D.)) |
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Subject Keywords: | elliptic,semiplane,design,graph,matrix,cage,code,error,correcting,cyclic |
Degree Grantor: | California Institute of Technology |
Division: | Physics, Mathematics and Astronomy |
Major Option: | Mathematics |
Thesis Availability: | Public (worldwide access) |
Research Advisor(s): |
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Thesis Committee: |
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Defense Date: | 24 September 2009 |
Non-Caltech Author Email: | schroederb (AT) gmail.com |
Record Number: | CaltechTHESIS:10262009-141148765 |
Persistent URL: | https://resolver.caltech.edu/CaltechTHESIS:10262009-141148765 |
DOI: | 10.7907/84VV-S966 |
Default Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. |
ID Code: | 5330 |
Collection: | CaltechTHESIS |
Deposited By: | Brian Schroeder |
Deposited On: | 21 Dec 2009 18:46 |
Last Modified: | 08 Nov 2019 18:07 |
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