Two 1998 papers [ ISBN 3-540-64382-6, pp. 1-31, and ISBN 3-85366-890-9, pp. 1-29 ] of Itshak Borosh and myself outline the general theory of codes. It is a set-theoretic formulation which embraces essentially all known codes. The idea is that the composition d e consisting of the decode relation d following the encode relation e must never lead away from the plaintext symbol which was originally encoded. In other words the composite relation d e must be an identity partial function whose domain is a subset of the set of all plaintext symbols.

This extremely economical definition is framed to avoid any mention of origin, application, content, context, phenomenology or finiteness. It is meant to be a theory of all codes, and so must not exhibit features distinctive of any one type of code, however important it may be. Yet this general theory contains many results. Also, it leads naturally to several topics in group theory, graph theory, Booolean matrix theory and other areas. Perhaps its most significant aspect so far is the fact that a code is an object within universal algebra. And there are very natural homomorphisms of codes which cause the general theory of codes to obey the three standard isomorphism theorems of universal algebra.

The talk will capsule the two papers. It will also discuss further progress in the theory, as well as in applications of the theory to cryptography, error control, biology and other fields.

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