On the trifference problem for linear codes

TL;DR

This paper establishes an upper bound on the dimension of perfect 3-hash linear codes over 3, showing they cannot be too large, which advances understanding of their structural limitations.

Contribution

It proves a new upper bound on the dimension of perfect 3-hash linear codes in finite fields, highlighting a fundamental limitation.

Findings

Perfect 3-hash linear codes in 3 have dimension at most 4 n for some 5 4 > 0

The result constrains the possible parameters of such codes

Advances theoretical understanding of code limitations

Abstract

We prove that perfect $3$-hash linear codes in $\mathbb{F}_{3}^{n}$ must have dimension at most $ \left(\frac{1}{4}-\epsilon\right)n$ for some absolute constant $\epsilon > 0$.