Random vectors over finite fields

The study of random objects is a useful one in many applications and areas of mathematics. The Probabilistic Method, introduced by Paul Erdős and his many collaborators, was first used to study the behavior of random graphs and later to study properties of random objects. It has developed as a powerful tool in combinatorics as well as finding applications in linear algebra, number theory, and many other areas. In this dissertation, we will consider random vectors, in particular, dependency among random vectors. We will randomly choose vectors according to a specified probability distribution. We wish to determine how many vectors must be generated before the vectors are almost surely dependent, that is, there is a high probability that a subset of the vectors is linearly dependent. In Chapter 1, we will review previous work done in this area. A typical result in the study of random objects is a threshold function that describes the behavior of a given property of the objects. We will discuss previous threshold functions and methods used to find them. The results found for this problem before now have been for vectors of bounded or fixed weight. In Chapter 2, we will develop the methods we will use later on vectors of fixed weight. We will then use these methods in Chapter 3 to vary the probability model under which the vectors are generated. Instead of considering vectors of fixed weight, we will consider a general probability model for choosing the vectors: each position in a vector will be assigned a probability of containing a nonzero entry. Finally, in Chapter 4 we will specify a function for this probability. We will then find a threshold result for the specified probability model. This result will give a lower bound for the number of vectors needed before they are almost surely dependent. ACKNOWLEDGMENTS I would like to thank my advisor, Neil Calkin, for spending his time and energy to listen to and teach me. He has introduced me to countless ideas and given me many opportunities that have enabled me to develop as a researcher and a teacher. His never-ending ideas have taught me how to always have another question. I also thank my committee for taking their time to talk with me and read my work. I thank Kevin James for being an excellent example in the classroom and always taking the time to talk to me when I had questions. I also thank Gretchen Matthews and Wayne Goddard for their questions and suggestions that have already given me new ideas to work on. Jim Peterson also deserves a great deal of thanks for being a wonderful teacher and encourager during my time here. My time at Clemson would not be the same without the other graduate students and all the friends I have made here. They have always provided me with an outlet for ideas and questions; I have learned a great deal of math from the other students here. Office chats, card games, and frisbee on Fridays have kept me sane and I will remember these things fondly. I owe my family a great deal for their support and prayers. They have always encouraged me, listened to me, and if possible, are more excited than I am that I’ve finished this chapter of my life. They have loved me unconditionally and I am honored to make them proud. I also thank my husband for being an amazing support system for me as well. He has prayed with me, listened to my frustrations, celebrated my triumphs, encouraged me to continue when I didn’t know how, and has been a wonderful example. He is an inspiration to me. Finally, I would like to thank God, through whom all things are possible. TABLE OF CONTENTS Page TITLE PAGE . i ABSTRACT . iii ACKNOWLEDGMENTS . v LIST OF FIGURES . ix CHAPTER 1. Random Vector History . 1 2. More on Fixed Weight Vectors . 13 2.