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On symmetric 2-(70,24,8) designs with an automorphism of order 6

Sanja Rukavina, Vladimir D. Tonchev

Abstract

In this paper we analyze possible actions of an automorphism of order six on a $2$-$(70, 24, 8)$ design, and give a complete classification for the action of the cyclic automorphism group of order six $G= \langle ρ\rangle \cong Z_6 \cong Z_2 \times Z_3$ where $ρ^3$ fixes exactly $14$ points (blocks) and $ρ^2$ fixes $4$ points (blocks). Up to isomorphism, there are $3718$ such designs. This result significantly increases the number of known $2$-$(70,24,8)$ designs.

On symmetric 2-(70,24,8) designs with an automorphism of order 6

Abstract

In this paper we analyze possible actions of an automorphism of order six on a - design, and give a complete classification for the action of the cyclic automorphism group of order six where fixes exactly points (blocks) and fixes points (blocks). Up to isomorphism, there are such designs. This result significantly increases the number of known - designs.
Paper Structure (5 sections, 7 theorems, 2 equations, 4 tables)

This paper contains 5 sections, 7 theorems, 2 equations, 4 tables.

Table of Contents

  1. Introduction
  2. A construction of $2$-$(70,24,8)$ designs with an automorphism of order six
  3. Possible actions of an automorphism of order six on a $2$-$(70,24,8)$ design
  4. New symmetric $2$-$(70,24,8)$ designs
  5. On the binary codes of $2$-$(70,24,8)$ designs

Key Result

Proposition 2.1

ord9 Let $p$ and $q$ be two distinc prime numbers and $G= \langle \rho \rangle \cong Z_{p \cdot q} \cong Z_p \times Z_q$ be a cyclic automorphism group of a symmetric block design $\mathcal{D}$, Then the $G$-orbits of points (or blocks) of the design $\mathcal{D}$ having length $p$ or $q$ consist of

Theorems & Definitions (8)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Remark 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 2.8