Symmetric 2$-$(36,15,6) designs with an automorphism of order two
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| Format: | Preprint |
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2024
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| _version_ | 1866912312080728064 |
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| author | Rukavina, Sanja Tonchev, Vladimir D. |
| author_facet | Rukavina, Sanja Tonchev, Vladimir D. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_03381 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Symmetric 2$-$(36,15,6) designs with an automorphism of order two Rukavina, Sanja Tonchev, Vladimir D. Combinatorics 05B05, 94B05 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. |
| title | Symmetric 2$-$(36,15,6) designs with an automorphism of order two |
| topic | Combinatorics 05B05, 94B05 |
| url | https://arxiv.org/abs/2403.03381 |