Modern data systems are commonly studied through computational and information-theoretic methods, while their algebraic properties remain largely unexplored. This paper introduces a mathematical framework for modelling data systems using algebraic structures drawn from group theory and coding theory. The central result is an axiomatic classifier: the coding-theoretic capability of a finite algebraic structure (group, ring, or field) is shown to be determined by its axiom signature (the set of algebraic axioms it satisfies), with each axiom acting as a gate that enables a specific capability (inverses enable the algebraic Hamming metric; commutativity enables syndrome decoding via quotient groups; field structure enables MDS codes via polynomial evaluation). Three types of results are presented. Proved: the axiomatic classification theorem; that ACID transactions form a monoid under sequential composition (not a group); that schema-preserving transformations form a finite group whose orbits, counted by Burnside's lemma, yield exact deduplication of equivalent configurations; and that every integrity-preserving bijection of a data system defines an algebraizable equivalence class under a finite group action. Demonstrated: explicit code constructions over $\mathbb{F}_5$, $\mathbb{Z}_4$ (yielding codes inaccessible to classical $\mathbb{F}_q$-linear theory), and $(2^U, \triangle)$ (yielding group-theoretic anomaly detection for set-valued data). Proposed: an encoding efficiency measure Ef and an optimization functional $Φ$ with tunable weights, whose induced ranking is verified computationally to be consistent with the axiomatic classification for the structures studied.

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