Improved Upper Bound for the Size of a Trifferent Code

A subset
$\mathcal{C} \subseteq\{0,1,2\}^n$
is said to be a 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,
$\{x(i), y(i), z(i)\}=\{0,1,2\}$
. 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 [Eli88], and is a longstanding open problem in the area. Elias had shown that
$T(n) \leq 2 \times(3 / 2)^n$
and prior to our work the best upper bound was
$T(n) \leq 0.6937 \times(3 / 2)^n$
due to Kurz [Kur24]. We improve this bound to
$T(n) \leq c \times n^{-2 / 5} \times(3 / 2)^n$
where
$c$
is an absolute constant.