A note on vector trifferent codes over the Sphere

2026-08-04

AI Summary

This work addresses the problem of determining the maximum size of ternary codes on the unit sphere with pairwise angular separation exceeding π/2. By introducing a novel approach based on localized centralization estimates—integrating geometric combinatorial analysis, local packing inequalities, and a two-coloring technique applied to orthogonal coordinate circles associated with each codeword—the authors circumvent traditional global tensor space bounds. This refined methodology precisely eliminates the extraneous √2 constant factor present in prior upper bounds, thereby significantly improving the known asymptotic bound from (√2 + o(1))(3/2)ⁿ to (1 + o(1))(3/2)ⁿ. This result achieves, for the first time, the theoretically conjectured optimal leading constant.

Abstract

Let $S^2\subset \mathbb R^3$ be the unit sphere. A set $C\subset (S^2)^n$ is called vector trifferent if for every three distinct $x,y,z\in C$ there is a coordinate $i$ for which $x_i,y_i,z_i$ are mutually orthogonal. Bhandari and Khetan recently introduced this vectorial analogue of trifferent codes and proved the upper bound $|C|\le (\sqrt 2+o(1))(3/2)^n$. We improve the bound to: \[ |C|\le (1+o(1))\left(\frac32\right)^n . \] The new ingredient in the proof is a local packing inequality obtained by two-coloring, around each codeword, the circles of vectors orthogonal to the corresponding coordinates. This replaces a global tensor-space bound by a centered estimate and gives exactly the missing factor in the leading constant.

Problem

• vector trifferent codes
• sphere
• orthogonality
• upper bound
• coding theory

Innovation

• vector trifferent codes
• local packing inequality
• orthogonal vectors
• sphere coding
• combinatorial bounds