Home / Current Issue / Paper 1723031
Finite Modular Coordinate Geometry and Algebraic Structures of Hypercubes
Subject area: Science,Engineering and Technology · Area of research: Mathematics
DOI: 10.64388/IREV10I3-1723031
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.
References
[1] Harary, F. (1969). Graph Theory. Addison-Wesley, Reading, Massachusetts.
[2] West, D. B. (2001). Introduction to Graph Theory (2nd ed.). Prentice Hall.
[3] MacWilliams, F. J., & Sloane, N. J. A. (1977). The Theory of Error-Correcting Codes. North-Holland.
[4] Roman, S. (1992). Coding and Information Theory. Springer.
[5] Lidl, R., & Niederreiter, H. (1997). Finite Fields (2nd ed.). Cambridge University Press. Crossref
[6] Dummit, D. S., & Foote, R. M. (2004). Abstract Algebra (3rd ed.). Wiley.
[7] Hall, M. Jr. (1959). The Theory of Groups. Macmillan.
[8] Biggs, N. (1993). Algebraic Graph Theory (2nd ed.). Cambridge University Press. Crossref
[9] Nielsen, M. A., & Chuang, I. L. (2010). Quantum Computation and Quantum Information (10th Anniversary ed.). Cambridge University Press.
[10] Kaye, P., Laflamme, R., & Mosca, M. (2007). An Introduction to Quantum Computing. Oxford University Press.
[11] van Lint, J. H. (1999). Introduction to Coding Theory (3rd ed.). Springer. Crossref
[12] Beth, T., Jungnickel, D., & Lenz, H. (1999). Design Theory (2nd ed.). Cambridge University Press. Crossref
[13] Pilz, G. (1977). Near-Rings: The Theory and Its Applications. North-Holland.
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},
doi = {https://doi.org/10.64388/IREV10I3-1723031}
}