A general theory of codes, II: Paradigms and homomorphisms

Abstract

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.