Multidimensional Convolutional Codes Dedication

Acknowledgements I would like to thank many people who helped and encouraged me with this work. My advisor, Joachim Rosenthal, worked with me long and patiently so that I could produce this dissertation. My committee members, Heide Gl using-L uerren, Alan Howard, and Juan Migliore, made many suggestions that have improved this work greatly. My academic \siblings", Brian Allen, Roxana Smarandache, and Eric York helped me in many ways, from technical assistance with my computer to explanations of some of the ner points of coding theory, but most importantly with their friendship. The faculty, staa, and graduate student body of the Notre Dame Department of Mathematics made working toward my Ph.D. a great experience. Chris Peterson gave me multiple explanations of various topics in algebraic geometry and commutative algebra. M. S. Ravi pointed me in the right direction regarding some diierences between 2-dimensional and higher-dimensional codes. Conversations with Philippe Loustaunau got me started thinking about ways of calculating the distance of a multidimensional convolutional code. Saint Mary's University of Minnesota gave me both nancial support and suucient time to pursue graduate studies. I received nancial support from the University of Notre Dame through the Mathematics Department and through a fellowship from the Center for Applied Mathematics. I also received funding from the NSF through my advisor's grant (DMS-96-10389). Finally without the love, support, and patience of my family this project would have been totally beyond my grasp. Abstract In this dissertation some of the theory of multidimensional convolutional codes is developed. Given a nite eld, F, consider the polynomial ring R = Fz 1 ; :::; z m ] in m indeterminates over F. An m-dimensional convolutional code of length n is an R-submodule of R n. With this deenition, many results and techniques from commutative algebra and algebraic geometry are available for the study of m-dimensional convolutional codes. These connections are explored in this dissertation (especially in Chapter 3). Much attention in the literature has been given to 1-dimensional convolutional codes, a slight amount to 2-dimensional convolutional codes, and almost none to higher-dimensional convolutional codes. It is worth noting that m-dimensional convolutional codes are a nontrivial generalization of the 1-dimensional case. There are interesting diierences between the 1-and 2-dimensional cases, and again between the 2-and 3-dimensional cases. Many of these diierences are considered in this work. A code construction considered in Chapter 4 gives a family of codes for …