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Trifferent codes with small lengths

Sascha Kurz

Abstract

A code $C \subseteq \{0, 1, 2\}^n$ of length $n$ is called trifferent if for any three distinct elements of $C$ there exists a coordinate in which they all differ. By $T(n)$ we denote the maximum cardinality of trifferent codes with length. $T(5)=10$ and $T(6)=13$ were recently determined. Here we determine $T(7)=16$, $T(8)=20$, and $T(9)=27$. For the latter case $n=9$ there also exist linear codes attaining the maximum possible cardinality $27$.

Trifferent codes with small lengths

Abstract

A code of length is called trifferent if for any three distinct elements of there exists a coordinate in which they all differ. By we denote the maximum cardinality of trifferent codes with length. and were recently determined. Here we determine , , and . For the latter case there also exist linear codes attaining the maximum possible cardinality .
Paper Structure (6 sections, 6 theorems, 15 equations, 3 tables)

This paper contains 6 sections, 6 theorems, 15 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Results obtained by exhaustive enumeration
  3. Upper bounds involving the minimum Hamming distance
  4. Upper bounds involving the number of occurrences of symbols
  5. Linear trifferent codes
  6. Trifferent codes with small lengths and relatively large cardinalities

Key Result

Lemma 1

Let $C$ be a trifferent code with length $n$ and $c\in C$ be arbitrary. Then, is a trifferent code with length $n+1$, minimum Hamming distance $1$, and cardinality $\# C+1$.

Theorems & Definitions (11)

  • Lemma 1
  • proof
  • Corollary 2
  • Theorem 3
  • proof
  • Lemma 4
  • proof
  • Corollary 5
  • proof
  • Theorem 6
  • ...and 1 more