Combinatorial problems related to codes, designs and finite geometries
Başlık çevirisi mevcut değil.
- Tez No: 403313
- Danışmanlar: Dr. VLADIMIR D. TONCHEV
- Tez Türü: Doktora
- Konular: Matematik, Mathematics
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 2017
- Dil: İngilizce
- Üniversite: Michigan Technological University
- Enstitü: Yurtdışı Enstitü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 206
Özet
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Özet (Çeviri)
An investigation of an open case of the famous conjecture made by Hamada [30] is carried out in the rst part of this dissertation. In 1973, Hamada made the following conjecture: Let D be a geometric design having as blocks the d-subspaces of PG(n; q) or AG(n; q), and let m be the p-rank of D. If D0 is a design with the same parameters as D, then the p-rank of D0 is greater or equal to m, and equality holds if and only if D0 is isomorphic to D. In 1986, Tonchev [73], and more recently Harada, Lam and Tonchev [41], Jungnickel and Tonchev [48], and Clark, Jungnickel and Tonchev [15] found designs having the same parameters and p-rank as certain geometric designs, hence providing counter-examples to the \only-if" part of Hamada's conjecture. We discuss some properties of the three known nonisomorphic 2-(64,16,5) designs of 2- rank 16, one being the design of the planes in the 3-dimensional ane geometry over the eld of order 4. We also try to nd an algebraic way to use the similarities between these designs in a search for counter-examples to Hamada's conjecture in ane spaces of higher dimension. Currently we know the existence of 22 projective planes of order 16 up to isomorphism, of which 4 are self dual. In the second part of this thesis, details of 2-(52,4,1) designs associated with known maximal 52-arcs are provided. A number of new maximal (52,4)-arcs in two of the known projective planes of order 16 are established. Newly discovered maximal (52,4)-arcs give new connections between the projective planes of order 16 as well. Previously the number of pairwise non-isomorphic resolutions of 2-(52,4,1) designs was 30 [20]. With the results in Tables 4.2 and 4.3, this bound is improved. Details of partial geometries coming from known maximal 52- arcs (including ours) are summarized in Table 4.4. It was pointed out that 104-sets of type (4,8) might arise from the unions of two disjoint maximal (52,4)-arcs. We detail the discovery of 37 new 104-sets of type (4,8), 18 of which come from unions of non-isomorphic maximal 52-arcs. Previous to our work, no such examples were known to exist. We discovered that the Johnson plane also contains disjoint maximal 52-arcs. Previously no such sets in the Johnson plane were known.
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