Improved Upper Bound for the Size of a Trifferent Code

A subset <tex>$\mathcal{C} \subseteq\{0,1,2\}^n$</tex> is said to be a trifferent code (of block length <tex>$n$</tex>) if for every three distinct codewords <tex>$x, y, z \in \mathcal{C}$</tex>, there is a coordinate <tex>$i \in\{1,2, \ldots, n\}$</tex> where they all differ, that is, <tex>$\{x(i), y(i), z(i)\}=\{0,1,2\}$</tex>. Let <tex>$T(n)$</tex> denote the size of the largest trifferent code of block length <tex>$n$</tex>. Understanding the asymptotic behavior of <tex>$T(n)$</tex> is closely related to determining the zero-error capacity of the (3/2)-channel defined by Elias [Eli88], and is a longstanding open problem in the area. Elias had shown that <tex>$T(n) \leq 2 \times(3 / 2)^n$</tex> and prior to our work the best upper bound was <tex>$T(n) \leq 0.6937 \times(3 / 2)^n$</tex> due to Kurz [Kur24]. We improve this bound to <tex>$T(n) \leq c \times n^{-2 / 5} \times(3 / 2)^n$</tex> where <tex>$c$</tex> is an absolute constant.