Finite Modular Coordinate Geometry and Algebraic Structures of Hypercubes

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 Cq,n=Z_q^n, with the binary case C2,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.