Unitals in projective planes of order 16
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Dosyalar
Tarih
2021
Yazarlar
Dergi Başlığı
Dergi ISSN
Cilt Başlığı
Yayıncı
Scientific Technical Research Council Turkey-Tubitak
Erişim Hakkı
info:eu-repo/semantics/openAccess
Özet
In this study, we perform computer searches for unitals in planes of order 16. The number of known nonisomorphic unitals in these planes is improved to be 261. Some data related to 2- (65, 5, 1) designs associated with unitals are given. New lower bounds on the number of unital designs in projective planes of order 16 and 2- (65, 5, 1) designs are established. The computations show that thirty-nine unitals can be embedded in two or more nonisomorphic projective planes of order 16. Fifteen new connections between planes of order 16 (based on unitals) are found. All unitals found by the algorithms used in this study are explicitly listed. We assume familiarity with the basic facts from combinatorial design theory and finite geometries [5, 9, 16]. A t-(v, k, ?) design (t-design) is a pair D = {X, B} of a set X of cardinality v, called points, and a collection B of k-subsets of X, called blocks, such that every t points appear together in exactly ? blocks. A 2-design with ? = 1 is called a Steiner design. The incidence matrix of a 2-(v, k, ?) design D is a matrix M = (mij) with rows labeled by the blocks of D, columns labeled by the points of D, where mi,j = 1 if the ith block contains the j th point and mi,j = 0 otherwise. For a prime p, the rank of the incidence matrix of design D over a finite field of characteristic p is called the p-rank of D. Two designs D and D? are called isomorphic if there is a bijection between their point sets that maps
Açıklama
Anahtar Kelimeler
Unital, projective plane, Steiner design, graph isomorphism, orbits, automorphism group, Buekenhout-Metz Unitals, Designs
Kaynak
Turkish Journal of Mathematics
WoS Q Değeri
Q3
Scopus Q Değeri
Q2
Cilt
45
Sayı
2
