A geometric approach to Mathon maximal arcs
نویسندگان
چکیده
In 1969, Denniston gave a construction of maximal arcs of degree d in Desarguesian projective planes of even order q, for all d dividing q. In 2002 Mathon gave a construction method generalizing the one of Denniston. We will give a new geometric approach to these maximal arcs. This will allow us to count the number of non-isomorphic Mathon maximal arcs of degree 8 in PG(2, 2h), h 6= 7 and prime. In GF(27) a new class of Mathon maximal arcs of degree 8 arises which admits a Singer group on the 7 conics of these arcs.
منابع مشابه
Partial flocks of the quadratic cone yielding Mathon maximal arcs
In [6] N. Hamilton and J. A. Thas describe a link between maximal arcs of Mathon type and partial flocks of the quadratic cone. This link is of a rather algebraic nature. In this paper we establish a geometric connection between these two structures. We also define a composition on the flock planes and use this to work out an analogue of the synthetic version of Mathon’s Theorem (see [3]). Fina...
متن کاملSinger 8-arcs of Mathon type in PG(2, 27)
In [3] De Clerck, De Winter and Maes counted the number of non-isomorphic Mathon maximal arcs of degree 8 in PG(2, 2), h 6= 7 and prime. In this article we will show that in PG(2, 2) a special class of Mathon maximal arcs of degree 8 arises which admits a Singer group (i.e. a sharply transitive group) on the 7 conics of these arcs. We will give a detailed description of these arcs, and then cou...
متن کاملMaximal Arcs in PG(2, q) and partial flocks of the quadratic cone
In this paper we show that there are several other structures that arise from the functions associated with the maximal arcs of Mathon type. So it is shown that maximal arcs of Mathon type are equivalent to additive partial flocks of the quadratic cone in PG(3, q) and to additive partial q-clans. Further they yield partial ovoids of Q(5, q), partial spreads of lines of PG(3, q), translation k-a...
متن کاملOn Mathon’s Construction of Maximal Arcs in Desarguesian
In a recent paper [M], Mathon gives a new construction of maximal arcs which generalizes the construction of Denniston. In relation to this construction, Mathon asks the question of determining the largest degree of a non-Denniston maximal arc arising from his new construction. In this paper, we give a nearly complete answer to this problem. Specifically, we prove that when m ≥ 5 and m 6= 9, th...
متن کاملOn Mathon’s construction of maximal arcs in Desarguesian planes
We study the problem of determining the largest d of a non-Denniston maximal arc of degree 2 generated by a fp; 1g-map in PGð2; 2Þ via a recent construction of Mathon [9]. On one hand, we show that there are fp; 1g-maps that generate non-Denniston maximal arcs of degree 2ðmþ1Þ=2, where md 5 is odd. Together with Mathon’s result [9] in the m even case, this shows that there are always fp; 1g-map...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- J. Comb. Theory, Ser. A
دوره 118 شماره
صفحات -
تاریخ انتشار 2011