A note on vector trifferent codes over the Sphere

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.