The generalized trifference problem

TL;DR

This paper investigates the generalized trifference problem, establishing bounds, phase transition thresholds, and explicit constructions for the maximum size of ternary vector sets with pairwise differences, extending classical open problems.

Contribution

It introduces bounds, phase transition analysis, and explicit constructions for the generalized trifference problem, connecting it to finite geometry and solving for small parameters.

Findings

Bounds on T(n, m) for various m

Identification of phase transition thresholds

Exact values for small parameters

Abstract

We study the problem of finding the largest number $T(n, m)$ of ternary vectors of length $n$ such that for any three distinct vectors there are at least $m$ coordinates where they pairwise differ. For $m = 1$, this is the classical trifference problem which is wide open. We prove upper and lower bounds on $T(n, m)$ for various ranges of the parameter $m$ and determine the phase transition threshold on $m=m(n)$ where $T(n, m)$ jumps from constant to exponential in $n$. By relating the linear version of this problem to a problem on blocking sets in finite geometry, we give explicit constructions and probabilistic lower bounds. We also compute the exact values of this function and its linear variation for small parameters.