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:10.7907/84VVS966. https://resolver.caltech.edu/CaltechTHESIS:10262009141148765
Abstract
An elliptic semiplane is a λfold of a symmetric 2(v,k,λ) design, where parallelism is transitive. We prove existence and uniqueness of a 3fold 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 m_{i}(x) be the minimal polynomial of ω^{i}. The code of length n = 2^{r}1 generated by m_{1}(x)m_{t}(x) is denoted C_{r}^{t}. We give a recursive formula for the number of codewords of weight 4 in C_{r}^{11}r for each r.
Item Type:  Thesis (Dissertation (Ph.D.)) 

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): 

Thesis Committee: 

Defense Date:  24 September 2009 
NonCaltech Author Email:  schroederb (AT) gmail.com 
Record Number:  CaltechTHESIS:10262009141148765 
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:10262009141148765 
DOI:  10.7907/84VVS966 
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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