Contributions to a General Theory of Codes
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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.
Holcomb, Trae (2003). Contributions to a General Theory of Codes. Doctoral dissertation, Texas A&M University. Texas A&M University. Available electronically from