Nets of order 4m+2: linear dependence and dimensions of codes




Howard, Leah

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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.



combinatorics, design theory