Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Improved Upper Bound for the Size of a Trifferent Code

Siddharth Bhandari, Abhishek Khetan

TL;DR

The paper advances the trifference problem by improving the upper bound on the largest trifferent code size $T(n)$ from the classical $2\times (3/2)^n$ to $T(n)\le c\times n^{-2/5}\times (3/2)^n$, tying the bound to the study of $r$-bounded trifferent codes and their densities. It introduces $T_b(n,r)$ and the density $\rho_b(n,r)$, and derives new upper bounds for small $r$ using a graph-theoretic approach and the Kővári--Sós--Turán theorem, leading to $\rho_b(n,3)\le c\times n^{-2/5}\times 2^{-n}$ and, consequently, the main $T(n)$ bound. Additionally, the work provides lower-bound constructions and growth analyses for $r$-bounded codes, including exact results for $r=1$ and recursive constructions for powers of $3$, and introduces the deficit function $\Delta_r$ to understand asymptotics as $r$ grows. These contributions shed light on the zero-error capacity of the $(3/2)$-channel and lay groundwork for potential refinements for larger $q$ and tighter asymptotics.

Abstract

A subset $\mathcal{C}\subseteq\{0,1,2\}^n$ is said to be a $\textit{trifferent}$ 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'88, and is a long-standing 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'23. We improve this bound to $T(n)\leq c \times n^{-2/5}\times (3/2)^n$ where $c$ is an absolute constant.

Improved Upper Bound for the Size of a Trifferent Code

TL;DR

The paper advances the trifference problem by improving the upper bound on the largest trifferent code size from the classical to , tying the bound to the study of -bounded trifferent codes and their densities. It introduces and the density , and derives new upper bounds for small using a graph-theoretic approach and the Kővári--Sós--Turán theorem, leading to and, consequently, the main bound. Additionally, the work provides lower-bound constructions and growth analyses for -bounded codes, including exact results for and recursive constructions for powers of , and introduces the deficit function to understand asymptotics as grows. These contributions shed light on the zero-error capacity of the -channel and lay groundwork for potential refinements for larger and tighter asymptotics.

Abstract

A subset is said to be a code (of block length ) if for every three distinct codewords , there is a coordinate where they all differ, that is, is same as . Let denote the size of the largest trifferent code of block length . Understanding the asymptotic behavior of is closely related to determining the zero-error capacity of the -channel defined by Elias'88, and is a long-standing open problem in the area. Elias had shown that and prior to our work the best upper bound was due to Kurz'23. We improve this bound to where is an absolute constant.
Paper Structure (3 sections, 11 theorems, 16 equations, 2 figures)

This paper contains 3 sections, 11 theorems, 16 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Upper bounds on the size of $r$-bounded trifferent codes
  3. Lower Bounds on the size of $r$-bounded trifferent codes

Key Result

Theorem 1.1

There exists a universal constant $c$ with the following property. Let $\mathcal{C}\subseteq \{0,1,2\}^n$ be a trifferent code of block length $n$ as defined in defn:qhashcode. Then, $|\mathcal{C}| \leq c \times n^{-2/5}\times (3/2)^n$. Thus, $T(n)\leq c \times n^{-2/5}\times (3/2)^n$.

Figures (2)

  • Figure 1: Illustrating the proof for $r = 2$. The collection of codewords before the application of the PHP is displayed in the left column of the figure: they correspond to the edges of the $K_{3,9}$ supposed to exist for the sake of contradiction. For example, the codeword $x_{3,2}$ corresponds to the highlighted edge $\{i_3,j_2\}$, and the first block of codewords on the left corresponds to the edges incident to $i_1$. After applying the PHP we are left with the non-grayed codewords in the right column corresponding to the set $T'=\{j_1,j_2\}$. Eventually, we find three codewords from the non-grayed ones which do not exhibit the trifference property.
  • Figure 2: The case when there are only two points and two lines.

Theorems & Definitions (27)

  • Definition 1.1: $q$-perfect hash codes & Trifferent codes
  • Theorem 1.1: Main theorem
  • Definition 1.2: Rate & Capacity
  • Remark 1.2
  • Theorem 1.2: Main theorem
  • Definition 1.3: $r$-bounded trifferent codes
  • Lemma 1.3: Size of trifferent codes in terms of $r$-bounded trifferent codes
  • proof
  • Remark 1.4
  • Definition 1.4: $r$-bounded density
  • ...and 17 more