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Symmetric 2$-$(36,15,6) designs with an automorphism of order two

Sanja Rukavina, Vladimir D. Tonchev

TL;DR

This work delivers a complete classification of symmetric $2$-$(36,15,6)$ designs admitting an automorphism of order two by exploiting an orbit-matrix framework and exhaustive computational enumeration across all fixed-point scenarios. It combines orbit-type analysis with GAP and the DESIGN package to enumerate nonisomorphic designs and determine self-duality across cases. The results include a total of $1 547 701$ designs with such an automorphism, of which $135 779$ are self-dual, and they yield near-extremal ternary self-dual codes of length $36$ (with $A_9=104$ or $A_9=120$) alongside the known extremal code $C(17)$. The Appendix provides explicit incidence matrices enabling the construction and verification of these near-extremal codes, reinforcing the bridge between combinatorial designs and ternary coding theory.

Abstract

The parameters 2-(36,15,6) are the smallest parameters of symmetric designs for which a complete classification up to isomorphism is yet unknown. Bouyukliev, Fack and Winne classified all 2-$(36,15,6)$ designs that admit an automorphism of odd prime order, and gave a partial classification of such designs that admit an automorphism of order 2. In this paper, we give the classification of all symmetric 2-$(36,15,6)$ designs that admit an automorphism of order two. It is shown that there are exactly $1 547 701$ nonisomorphic such designs, $135 779$ of which are self-dual designs. The ternary linear codes spanned by the incidence matrices of these designs are computed. Among these codes, there are near-extremal self-dual codes with previously unknown weight distributions.

Symmetric 2$-$(36,15,6) designs with an automorphism of order two

TL;DR

This work delivers a complete classification of symmetric - designs admitting an automorphism of order two by exploiting an orbit-matrix framework and exhaustive computational enumeration across all fixed-point scenarios. It combines orbit-type analysis with GAP and the DESIGN package to enumerate nonisomorphic designs and determine self-duality across cases. The results include a total of designs with such an automorphism, of which are self-dual, and they yield near-extremal ternary self-dual codes of length (with or ) alongside the known extremal code . The Appendix provides explicit incidence matrices enabling the construction and verification of these near-extremal codes, reinforcing the bridge between combinatorial designs and ternary coding theory.

Abstract

The parameters 2-(36,15,6) are the smallest parameters of symmetric designs for which a complete classification up to isomorphism is yet unknown. Bouyukliev, Fack and Winne classified all 2- designs that admit an automorphism of odd prime order, and gave a partial classification of such designs that admit an automorphism of order 2. In this paper, we give the classification of all symmetric 2- designs that admit an automorphism of order two. It is shown that there are exactly nonisomorphic such designs, of which are self-dual designs. The ternary linear codes spanned by the incidence matrices of these designs are computed. Among these codes, there are near-extremal self-dual codes with previously unknown weight distributions.
Paper Structure (14 sections, 2 theorems, 3 equations, 12 tables)

This paper contains 14 sections, 2 theorems, 3 equations, 12 tables.

Table of Contents

  1. Introduction
  2. Classification of symmetric $2$-$(36,15,6)$ designs with an automorphism of order two
  3. Symmetric $2$-$(36,15,6)$ designs admitting an automorphism of order two without fixed points
  4. Symmetric $2$-$(36,15,6)$ designs admitting an automorphism of order two acting with four fixed points
  5. Symmetric $2$-$(36,15,6)$ designs with an automorphism of order two fixing six points
  6. Symmetric $2$-$(36,15,6)$ designs with an automorphism of order two fixing eight points
  7. Symmetric $2$-$(36,15,6)$ designs with an automorphism of order two fixing ten points
  8. Symmetric $2$-$(36,15,6)$ designs with an automorphism of order two fixing $12$ points
  9. Symmetric $2$-$(36,15,6)$ designs with an automorphism of order two fixing $14$ points
  10. Symmetric $2$-$(36,15,6)$ designs with an automorphism of order two fixing $16$ points
  11. Symmetric $2$-$(36,15,6)$ designs with an automorphism of order two fixing $18$ points
  12. Enumeration of symmetric $2$-$(36,15,6)$ designs with an automorphism of order two
  13. Near-extremal ternary self-dual codes of length $36$ related to the symmetric $2$-$(36,15,6)$ designs admitting an automorphism of order two
  14. Appendix

Key Result

Theorem 2.1

An automorphism of order two acts on a symmetric $2$-$(36,15,6)$ design with $f$ fixed points, where $f \in \{0,4,6,8,10,12,16 \}$. Up to isomorphism, there are $1 547 701$ symmetric $2$-$(36,15,6)$ designs admitting an automorphism of order two with the full automorphism groups as given in Table ta

Theorems & Definitions (2)

  • Theorem 2.1
  • Proposition 3.1