The generalized trifference problem

2025-05-12

AI Summary

This paper studies the maximum size $ T(n,m) $ of a set of ternary vectors of length $ n $, subject to the condition that any three distinct vectors differ pairwise in at least $ m $ coordinates. This generalizes the classical open “trifference” problem (corresponding to $ m = 1 $). The authors employ a synthesis of extremal combinatorics, finite geometry—particularly blocking set theory in projective planes—probabilistic methods, and explicit algebraic constructions. Their main contribution is the first precise characterization of the phase-transition threshold $ m = m(n) $: below this curve, $ T(n,m) $ remains bounded; above it, $ T(n,m) $ grows exponentially in $ n $. They further establish tight multi-segment upper and lower bounds, compute exact values for small parameters and linear variants, and uncover a deep connection to blocking set problems in finite projective geometry.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Determine maximum ternary vectors with pairwise differences
• Find phase transition threshold for exponential growth
• Relate linear problem to blocking sets in geometry

Innovation

• Upper and lower bounds on ternary vectors
• Phase transition threshold determination
• Blocking sets in finite geometry