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| Format: | Preprint |
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2024
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| Online-Zugang: | https://arxiv.org/abs/2403.03189 |
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| _version_ | 1866929264990879744 |
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| author | Huerta, Zazil Santizo Keranen, Melissa Tonchev, Vladimir |
| author_facet | Huerta, Zazil Santizo Keranen, Melissa Tonchev, Vladimir |
| contents | A maximal arc of degree k in a finite projective plane P of order q = ks is a set of (q-s+1)k points that meets every line of P in either k or 0 points. The collection of the nonempty intersections of a maximal arc with the lines of P is a resolvable Steiner 2-((q-s+1)k, k, 1) design. Necessary and sufficient conditions for a resolvable Steiner 2- design to be embeddable as a maximal arc in a projective plane were proved recently in [8]. Steiner designs associated with maximal arcs in the known projective planes of order 16 were analyzed in [6], where it was shown that some of the associated designs are embeddable in two non-isomorphic planes. Using MAGMA, we conducted an analysis to ascertain whether any of the 22 non-isomorphic 1-rotational 2-(52,4,1) designs, previously classified in [3], could be embedded in maximal arcs of degree 4 within projective planes of order 16. This paper presents a summary of our findings, revealing that precisely only one out of the the twenty-two 1-rotational designs from [3] is embeddable in a plane of order 16, being the Desarguesian plane P G(2, 16). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_03189 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Projective Planes of Order 16 Associated with 1-rotational 2-(52, 4, 1) Designs Huerta, Zazil Santizo Keranen, Melissa Tonchev, Vladimir Combinatorics A maximal arc of degree k in a finite projective plane P of order q = ks is a set of (q-s+1)k points that meets every line of P in either k or 0 points. The collection of the nonempty intersections of a maximal arc with the lines of P is a resolvable Steiner 2-((q-s+1)k, k, 1) design. Necessary and sufficient conditions for a resolvable Steiner 2- design to be embeddable as a maximal arc in a projective plane were proved recently in [8]. Steiner designs associated with maximal arcs in the known projective planes of order 16 were analyzed in [6], where it was shown that some of the associated designs are embeddable in two non-isomorphic planes. Using MAGMA, we conducted an analysis to ascertain whether any of the 22 non-isomorphic 1-rotational 2-(52,4,1) designs, previously classified in [3], could be embedded in maximal arcs of degree 4 within projective planes of order 16. This paper presents a summary of our findings, revealing that precisely only one out of the the twenty-two 1-rotational designs from [3] is embeddable in a plane of order 16, being the Desarguesian plane P G(2, 16). |
| title | On Projective Planes of Order 16 Associated with 1-rotational 2-(52, 4, 1) Designs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2403.03189 |