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Maximal cliques in the graph of $5$-ary simplex codes of dimension two

Mariusz Kwiatkowski, Andrzej Matraś, Mark Pankov, Adam Tyc

Abstract

We consider the induced subgraph of the corresponding Grassmann graph formed by $q$-ary simplex codes of dimension $2$, $q\ge 5$. This graph contains precisely two types of maximal cliques. If $q=5$, then for any two maximal cliques of the same type there is a monomial linear automorphism transferring one of them to the other. Examples concerning the cases $q=7,11$ finish the note.

Maximal cliques in the graph of $5$-ary simplex codes of dimension two

Abstract

We consider the induced subgraph of the corresponding Grassmann graph formed by -ary simplex codes of dimension , . This graph contains precisely two types of maximal cliques. If , then for any two maximal cliques of the same type there is a monomial linear automorphism transferring one of them to the other. Examples concerning the cases finish the note.
Paper Structure (8 sections, 11 theorems, 66 equations, 2 figures)

This paper contains 8 sections, 11 theorems, 66 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Simplex codes
  3. Main result
  4. Proof of Theorems \ref{['theorem-top-gen']} and \ref{['theorem-top-q5']}
  5. The inversion transformation of ${\mathcal{P}}(V)$
  6. Proof of Theorem \ref{['theorem-top-gen']}
  7. Proof of Theorem \ref{['theorem-top-q5']}
  8. Remarks on the case $q>5$

Key Result

Theorem 2

For every $q\ge 5$ the graph $\Gamma$ contains tops.

Figures (2)

  • Figure 1:
  • Figure 2:

Theorems & Definitions (27)

  • Remark 1
  • Theorem 2
  • Theorem 3
  • Lemma 4
  • proof
  • Remark 5
  • Proposition 6
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 17 more