Perfect Sphere Packing In The Boolean Space

Abstract

Perfect sphere packing in the Boolean space is a fundamental and complex problem with significant implications for coding theory, cryptography, and discrete mathematics. The classical solution to the perfect sphere packing problem was provided by Hamming via his well-known perfect codes. However, a major limitation of the traditional Hamming metric is its strict applicability, as it allows perfect partitioning only for spaces with specific, highly constrained dimensions. To address this structural limitation, this article introduces a novel distance metric specifically designed for Boolean hypercubes. The proposed metric modifies the topological properties of the space, making it mathematically viable to partition a Boolean space of any arbitrary dimension into disjoint, perfect spheres. We rigorously define the algebraic properties of this new distance function and demonstrate its consistency across various dimensions. Furthermore, we explore the structural characteristics of the resulting packings. This approach bypasses the classical dimensional constraints of Hamming codes, potentially opening new avenues for designing error-correcting codes and cryptographic primitives in non-traditional dimensions.