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

• Main Author: Schroeder, Brian Leroy
• Format: Others
• Published: 2010
• Online Access: https://thesis.library.caltech.edu/5330/1/paper.pdf; 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 <https://resolver.caltech.edu/CaltechTHESIS:10262009-141148765>

<p>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).</p> <p>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.</p> <p>Let ω be a primitive nth root of unity over GF(2), and let m_i(x) be the minimal polynomial of ωⁱ. The code of length n = 2^r-1 generated by m₁(x)m_t(x) is denoted C_r^t. We give a recursive formula for the number of codewords of weight 4 in C_r¹¹r for each r.</p>