Extremal ternary self-dual codes of length 36 and symmetric 2-(36,15,6) designs with an automorphism of order 2

TL;DR

This paper classifies symmetric 2-(36,15,6) designs with an automorphism of order 2 and shows their incidence matrices generate an extremal ternary self-dual code, specifically the Pless symmetry code.

Contribution

It provides a complete classification of such designs and establishes their connection to a known extremal ternary self-dual code.

Findings

Only one such design exists up to isomorphism.

The design's automorphism group has order 24.

The generated code is equivalent to the Pless symmetry code.

Abstract

In this note we report the classification of all symmetric 2-(36,15,6) designs that admit an automorphism of order 2 and their incidence matrices generate an extremal ternary self-dual code. It is shown that up to isomorphism, there exists only one such design, having a full automorphism group of order 24, and the ternary code spanned by its incidence matrix is equivalent to the Pless symmetry code.