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 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 (IEEE Trans Inform Theory 34(5):1070–1074, 1988), 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 (Example Counterexample 5:100139, 2024). We improve this bound to \(T(n)\le c \times n^{-2/5}\times (3/2)^n\) where c is an absolute constant.