Construction and Categories of Codes

. Blakley and Borosh introduced a general theory of codes, encompassing cryptographic and error control codes among others. They explored the properties of such general codes with methods from relational algebra and set theory. We provide a categorical point of view, which leads to new constructions of codes. We also exhibit a JordanH older type theorem and a Schreier refinement technique. 1 Introduction In the late twentieth century a vast proliferation of codes occurred. Many new cardinalities became common, especially large finite or infinite. Many new arithmetics -- infinite as well as finite -- could be found in the newly introduced arithmetic-based codes. Hilbert spaces are as integral to the theory of quantum error control as Hamming spaces to classical error control. But many new codes arose without arithmetic, amounting to mere codebooks or databases. Codes with no encode process, codes with no decode process, codes which encode every plaintext symbol into billions of di#eren..