Improved Upper Bound for the Size of a Trifferent Code

Abstract

A subset $\mathcal{C}\subseteq \{0,1,2{\}}^n$ is said to be a $\text{trifferent}$ code (of block length $n$) if for every three distinct codewords $x,y,z\in \mathcal{C}$, there is a coordinate $i\in \{1,2,\dots ,n\}$ where they all differ, that is, $\{x(i),y(i),z(i)\}$ is same as $\{0,1,2\}$. Let $T(n)$ denote the size of the largest trifferent code of block length $n$. Understanding the asymptotic behavior of $T(n)$ is closely related to determining the zero-error capacity of the $(3/2)$-channel defined by Elias'88, and is a long-standing open problem in the area. Elias had shown that $T(n)\le 2\times (3/2{)}^n$ and prior to our work the best upper bound was $T(n)\le 0.6937\times (3/2{)}^n$ due to Kurz'23. We improve this bound to $T(n)\le c\times n^{-2/5}\times (3/2{)}^n$ where $c$ is an absolute constant.

11 pages, 2 figures