An Algebraic Framework for Data Systems: Classification and Optimization of Algebraic Structures for Data Organization and Coding

@misc{inoa2026algebraic,
  title = {{An Algebraic Framework for Data Systems: Classification and Optimization of Algebraic Structures for Data Organization and Coding}},
  author = {Kendy Inoa},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.00273},
  eprint = {2609.00273},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.00273},
}

TY  - UNPB
TI  - An Algebraic Framework for Data Systems: Classification and Optimization of Algebraic Structures for Data Organization and Coding
AU  - Kendy Inoa
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2609.00273
UR  - https://arxiv.org/abs/2609.00273
AB  - 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 codes over finite groups, organized by an \emph{axiom signature}: each algebraic axiom of the data domain is shown to be \emph{sufficient} for a specific coding-theoretic capability, and each implication is given its precise reading (MDS parameters do not require a field; quotients require normality, not commutativity). Three types of results are presented. Proved: a Gilbert-Varshamov existence bound over arbitrary finite groups, commutative or not; explicit direct-sum codes over the elementary divisors of any finite abelian group; that transactions preserving a validity predicate form a submonoid of the full transformation monoid, which is not a group; that schema-preserving transformations form a finite group whose orbits are counted by Burnside's lemma; and that the constraint-preserving bijections of a data system form a finite group whose orbits partition the valid configurations. Demonstrated: an explicit code construction over 𝔽_5, with known codes over ℤ_4 and (2^U,△) cited to situate the framework. Proposed: an encoding efficiency measure Ef and an optimization functional Φ with tunable weights, both evaluated at declared coding instances. Weight-independent comparison holds under componentwise dominance; in general the weight simplex decomposes into polyhedral chambers on which the induced order is constant.
ER  -