Schroeder, Brian (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. http://resolver.caltech.edu/CaltechTHESIS:10262009-141148765
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 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^t_r. We give a recursive formula for the number of codewords of weight 4 in C^11_r for each r.
|Item Type:||Thesis (Dissertation (Ph.D.))|
|Degree Grantor:||California Institute of Technology|
|Division:||Physics, Mathematics and Astronomy|
|Thesis Availability:||Public (worldwide access)|
|Defense Date:||24 September 2009|
|Author Email:||schroederb (AT) gmail.com|
|Default Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Brian Schroeder|
|Deposited On:||21 Dec 2009 18:46|
|Last Modified:||26 Dec 2012 03:18|
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