Contributions to a General Theory of Codes

In 1997, Drs. G. R. Blakley and I. Borosh published two papers whose stated purpose was to present a general formulation of the notion of a code that depends only upon a code's structure and not its functionality. In doing so, they created a further generalization--the idea of a precode. Recently, Drs. Blakley, Borosh, and A. Klappenecker have worked on interpreting the structures and results in these pioneering papers within the framework of category theory. The purpose of this dissertation is to further the above work. In particular, we seek to accomplish the following tasks within the ``general theory of codes.' 1. Rewrite the original two papers in terms of the alternate representations of precodes as bipartite digraphs and Boolean matrices. 2. Count various types of bipartite graphs up to isomorphism, and count various classes of codes and precodes up to isomorphism. 3. Identify many of the classical objects and morphisms from category theory within the categories of codes and precodes. 4. Describe the various ways of constructing a code from a precode by ``splitting' the precode. Identify important properties of these constructions and their interrelationship. Discuss the properties of the constructed codes with regard to the factorization of homomorphisms through them, and discuss their relationship to the code constructed from the precode by ``smashing.' 5. Define a parametrization of a precode and give constructions of various parametrizations of a given precode, including a ``minimal' parametrization. 6. Use the computer algebra system, Maple, to represent and display a precode and its companion, opposite, smash, split, bald-split, and various parametrizations. Implement the formulae developed for counting bipartite graphs and precodes up to isomorphism

Developing a Mathematics Capstone Experience at The U.S. Air Force Academy

There are many details to consider when designing a capstone experience. Expectations of students, faculty workload, resources, logistics, and timing all play a part in the value of this endeavor. We discuss the experience of creating a research capstone experience from scratch including the components of our current course as well as lessons learned. Supplementary materials are available for this article. Go to the publisher’s online edition of PRIMUS to view the supplemental file

Developing a Mathematics Capstone Experience at The U.S. Air Force Academy

There are many details to consider when designing a capstone experience. Expectations of students, faculty workload, resources, logistics, and timing all play a part in the value of this endeavor. We discuss the experience of creating a research capstone experience from scratch including the components of our current course as well as lessons learned. Supplementary materials are available for this article. Go to the publisher’s online edition of PRIMUS to view the supplemental file