Nets of order 4m+2: linear dependence and dimensions of codes
Date
2009-08-24T18:55:43Z
Authors
Howard, Leah
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Abstract
A k-net of order n is an incidence structure consisting of n2
points and nk lines. Two lines are said to be parallel if they do not intersect.
A k-net of order n satisfies the following four axioms: (i) every line contains n
points; (ii) parallelism is an equivalence relation on the set of lines; (iii) there
are k parallel classes, each consisting of n lines and (iv) any two non-parallel
lines meet exactly once.
A Latin square of order n is an n by n array of symbols in which each
row and column contains each symbol exactly once. Two Latin squares L
and M are said to be orthogonal if the n2 ordered pairs (Li,j , Mi,j ) are all
distinct. A set of t mutual ly orthogonal Latin squares is a collection of Latin
squares, necessarily of the same order, that are pairwise orthogonal. A k-net
of order n is combinatorially equivalent to k − 2 mutually orthogonal Latin
squares of order n. It is this equivalence that motivates much of the work in
this thesis.
One of the most important open questions in the study of Latin squares
is: given an order n what is the maximum number of mutually orthogonal
Latin squares of that order? This is a particularly interesting question when
n is congruent to two modulo four. A code is constructed from a net by
defining the characteristic vectors of lines to be generators of the code over
the finite field F2 . Codes allow the structure of nets to be profitably explored
using techniques from linear algebra.
In this dissertation a framework is developed to study linear dependence
in the code of the net N6 of order ten. A complete classification and combinatorial description of such dependencies is given. This classification could
facilitate a computer search for a net or could be used in conjunction with
more refined techniques to rule out the existence of these nets combinatorially. In more generality relations in 4-nets of order congruent to two modulo
four are also characterized.
One type of dependency determined algebraically is shown not to be combinatorially feasible in a net N6 of order ten. Some dependencies are shown
to be related geometrically, allowing for a concise classification.
Using a modification of the dimension argument first introduced by
Dougherty [19] new upper bounds are established on the dimension of codes
of nets of order congruent to two modulo four. New lower bounds on some
of these dimensions are found using a combinatorial argument. Certain constraints on the dimension of a code of a net are shown to imply the existence
of specific combinatorial structures in the net.
The problem of packing points into lines in a prescribed way is related
to packing problems in graphs and more general packing problems in combinatorics. This dissertation exploits the geometry of nets and symmetry
of complete multipartite graphs and combinatorial designs to further unify
these concepts in the context of the problems studied here.
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Keywords
combinatorics, design theory