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On Elliptic Semiplanes, an Algebraic Problem in Matrix Theory, and Weight Enumeration of Certain Binary Cyclic Codes


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.


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.))
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):
  • Wilson, Richard M.
Thesis Committee:
  • Wilson, Richard M. (chair)
  • Ramakrishnan, Dinakar
  • Wales, David B.
Defense Date:24 September 2009
Non-Caltech Author Email:schroederb (AT)
Record Number:CaltechTHESIS:10262009-141148765
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:5330
Deposited By: Brian Schroeder
Deposited On:21 Dec 2009 18:46
Last Modified:08 Nov 2019 18:07

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