Improved Upper Bound for the Size of a Trifferent Code

Abstract: A subset $\mathcal{C}\subseteq{0,1,2}<sup>n$ is said to be a $\textit{trifferent}$ code (of block length $n$) if for every three distinct codewords $x,y, z \in \mathcal{C}$, there is a coordinate $i\in {1,2,\ldots,n}$ 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 $T(n)\leq 2\times (3/2)<sup>n$ and prior to our work the best upper bound was $T(n)\leq 0.6937 \times (3/2)<sup>n$ due to Kurz'23. We improve this bound to $$T(n)\leq c \times n^{-2/5}\times (3/2)<sup>n$$ where $c$ is an absolute constant.