A general theory of codes, II: Paradigms and homomorphisms
When two competing paradigms bear on a single area of study, investigators have more choices at their disposal. This is not always an advantage.
This paper, like its predecessor, adopts a paradigm for codes. This paradigm ignores the purposes which might have given rise to a code, the size of the code, or the arithmetic used in implementing the code. It concentrates solely on the (set-theoretic) structure of that code. Once adopted, this structure-oriented paradigm leads naturally to a theory of homomorphisms for the general theory of codes. Code homomorphisms satisfy the standard isomorphism theorems, respect certain important properties of codes, are compatible with products and quotients, and possess other desirable features. Thus, codes fit into general algebra alongside such familiar objects as groups, graphs and posets.
KeywordsEquivalence Relation General Algebra Dominant Paradigm Isomorphism Theorem Relation Isomorphism
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- [BLA89]G.R. Blakley and Catherine Meadows, Information theory without the finiteness assumption, III: Data compression and codes whose rates exceed unity, in H.J. Beker and E.C. Piper (Editors), Cryptography and Coding, Proceedings of the IMA Conference on Cryptography and Coding at the Royal Agricultural College, Cirencester, December, 1986, Clarendon Press, Oxford (1989), pp. 67–93.Google Scholar
- [BLA95]G.R. Blakley, Management of secret information: An abstract theory of codes, some implications for PKCs and secret sharing, and homomorphisms of relations and codes, in Proceedings, JAIST International Forum on Multimedia and Information Security, Japan Advanced Institute of Science and Technology, Tatsunokuchi, Ishikawa, Japan, 1–2 November (1995), pp. 55–83.Google Scholar
- [BLA96]G.R. Blakley and I. Borosh, Codes, in Pragocrypt, '96, Part 1, Proceedings of the First International Conference on the Theory and Applications of Cryptography, Pragocrypt '96, CTU Publishing House, Zikova 4, Prague 6, Czech Republic (1996), pp. 253–271.Google Scholar
- [BLA98]G.R. Blakley and I. Borosh, A general theory of codes, I: Basic concepts, in Contributions to General Algebra, 10, Proceedings of the Klagenfurt Conference on General Algebra, May 29–June 1, 1997, (1998) in press.Google Scholar
- [GOD68]R. Godement, Algebra, Hermann, Paris, and Houghton Mifflin, New York (1968).Google Scholar
- [GRA68]G. Grätzer, Universal Algebra, Van Nostrand, Princeton, New Jersey (1968).Google Scholar
- [KAH67]D. Kahn, The Codebreakers, Macmillan, New York (1967).Google Scholar
- [KOL56]A.N. Kolmogorov, On the Shannon Theory of Information in the Case of Continuous Signals, IEEE Transactions on Information Theory, vol. IT-2 (1956), pp. 102–108, Reprinted as pages 238–244 in D. Slepian (Editor) Key Papers in the Development of Information Theory, IEEE Press, New York (1974).Google Scholar
- [MAL73]A.I. Mal'cev, Algebraic Systems, Springer-Verlag, New York (1973).Google Scholar
- [MEA98]Catherine Meadows, Three paradigms in computer security, Proceedings of the 1997 New Security Paradigms Workshop, ACM (1998), in press.Google Scholar
- [PAL66]H. Paley and P.M. Weichsel, A First Course in Abstract Algebra, Holt, Rinehart and Winston, New York (1966).Google Scholar
- [TUF83]E.N. Tufte, The Visual Display of Quantitative Information, Graphics Press, Cheshire, Connecticut (1983).Google Scholar
- [TUF90]E.N. Tufte, Envisioning Information, Graphics Press, Cheshire, Connecticut (1990).Google Scholar
- [TUF96]E.N. Tufte, Visual Explanations: Images and Quantities, Evidence and Narrative, Graphics Press, Cheshire Connecticut (1996).Google Scholar