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Finite Modular Coordinate Geometry and Algebraic Structures of Hypercubes
Subject area: Science,Engineering and Technology · Area of research: Mathematics
Abstract
This paper develops a unified mathematical framework for finite modular coordinate spaces, beginning with the elementary but structurally rich relation between Z_2 and the vertices of a square, and extending it to cubes, n-dimensional hypercubes, ternary coordinate spaces, finite vector spaces, graph structures, coding theory, digital mathematics and quantum-information indexing. The principal coordinate object is C_{q,n}=Z_q^n, with the binary case C_{2,n}=F_2^n providing the vertex set of the n-dimensional hypercube. We establish results for vertex counts, adjacency, Hamming distance, Euclidean distance, Hamming spheres, faces, shortest paths and affine transformations. The q-ary extension gives a common language for binary and ternary coordinate systems. Applications are developed to binary digital images and voxels, error-correcting codes, hypercube computer networks, Boolean operations, and computational-basis indexing of qubits and qutrits. A mathematically precise bridge to quantum information is obtained through Pauli-X and generalized Weyl translation operators. We then propose a research programme for modular cubical algebra, in which graph-compatible operations, affine transformations and finite near-ring structures are studied together. The paper distinguishes established results from proposed research directions and identifies several problems suitable for further investigation.
Keywords
modular coordinate geometry; finite fields; F_2^n; F_3^n; hypercube; Hamming distance; coding theory; digital algebra; graph networks; Pauli-X; qutrits; modular transformations; near-rings.
How to cite this paper
@article{1723031,
author = {Dr. K V Rama Rao, Dr. Kondragunta Rama Krishnaiah},
title = {Finite Modular Coordinate Geometry and Algebraic Structures of Hypercubes},
journal = {Iconic Research And Engineering Journals},
year = {2026},
volume = {10},
number = {3},
pages = {1268-1276},
issn = {2456-8880},
url = {https://www.irejournals.com/formatedpaper/1723031.pdf},
abstract = {This paper develops a unified mathematical framework for finite modular coordinate spaces, beginning with the elementary but structurally rich relation between Z_2 and the vertices of a square, and extending it to cubes, n-dimensional hypercubes, ternary coordinate spaces, finite vector spaces, graph structures, coding theory, digital mathematics and quantum-information indexing. The principal coordinate object is C_{q,n}=Z_q^n, with the binary case C_{2,n}=F_2^n providing the vertex set of the n-dimensional hypercube. We establish results for vertex counts, adjacency, Hamming distance, Euclidean distance, Hamming spheres, faces, shortest paths and affine transformations. The q-ary extension gives a common language for binary and ternary coordinate systems. Applications are developed to binary digital images and voxels, error-correcting codes, hypercube computer networks, Boolean operations, and computational-basis indexing of qubits and qutrits. A mathematically precise bridge to quantum information is obtained through Pauli-X and generalized Weyl translation operators. We then propose a research programme for modular cubical algebra, in which graph-compatible operations, affine transformations and finite near-ring structures are studied together. The paper distinguishes established results from proposed research directions and identifies several problems suitable for further investigation.},
keywords = {modular coordinate geometry; finite fields; F_2^n; F_3^n; hypercube; Hamming distance; coding theory; digital algebra; graph networks; Pauli-X; qutrits; modular transformations; near-rings.},
month = {September},
}