scholarly journals (k,n,f) - Arcs in Galois Plane of Order Five

2006 ◽  
Vol 18 (3) ◽  
pp. 78-87
Author(s):  
Ban A. Q
Keyword(s):  
Galois Plane

scholarly journals On the Directions Determined by a Cartesian Product in an Affine Galois Plane

COMBINATORICA ◽  
2021 ◽  
Author(s):  
Daniel Di Benedetto ◽  
József Solymosi ◽  
Ethan P. White
Keyword(s):  
Cartesian Product ◽  
Galois Plane

The (11, 3)-arcs of the Galois plane of order 5

1973 ◽  
Vol 74 (2) ◽  
pp. 247-250 ◽  
Author(s):  
D. L. Bramwell ◽  
B. J. Wilson
Keyword(s):  
Electronic Computer ◽  
Geometrical Properties ◽  
Galois Plane ◽  
Image Position ◽  
Galois Planes ◽  
Low Order

1. It was shown by Barlotti(1) that the number, k, of points on a (k, n)-arc in a Galois plane S2, q, of order q, where n and q are coprime, satisfiesRegular arcs, in which all the points are of the same type have been studied by Basile and Brutti(2) and, for n = 3 by d'Orgeval(4). By means of an electronic computer Lunelli and Sce(5) have enumerated many arcs in Galois planes of low order. The object of this note is to show how the (11, 3)-arcs of S2, 5, none of which is regular, may be described using only geometrical properties.

The combinatorial structure of the Hughes plane

1970 ◽  
Vol 68 (2) ◽  
pp. 291-301 ◽  
Author(s):  
T. G. Room
Keyword(s):  
Cyclic Group ◽  
Singular Points ◽  
The Other ◽  
Combinatorial Structure ◽  
Combinatorial Properties ◽  
Cyclic Groups ◽  
Singer Cyclic Group ◽  
Galois Plane ◽  
Hughes Plane

AbstractThe main purpose of this paper is to describe the construction of an incidence table for the Hughes plane of order q2. To do this it is necessary first to construct a table of the same pattern for the Galois plane, and this requires the expression in terms of explicit matrices of the Singer cyclic group of the plane as the composition of cyclic groups of orders q2 + q + 1 and q2 − q + 1. The two tables constructed have some combinatorial properties of possible interest.In the Galois plane of order q2 there are two types of polarities, one with q2 + 1 singular points on a conic, and the other with q3 + 1 singular points on the Hermitian analogue of a conic. Correspondingly in the Hughes plane there are polarities with ½(q3 + q + 2) and with ½(q3 + 2q2 − q + 2) singular points.The paper concludes with the complete incidence table for the Hughes plane of order 25, together with the data necessary for its construction.The Hughes plane and some of its properties are described in Hughes(1), Zappa(2), Rosati(3) and Ostrom(4). The property which enables the Hughes plane to be most easily constructed from the corresponding Galois plane, and which will be used in this paper, is to be found in Room (5).

The combinatorial structure of the Hughes plane. II

1972 ◽  
Vol 72 (2) ◽  
pp. 135-139 ◽  
Author(s):  
T. G. Room
Keyword(s):  
Fixed Point ◽  
Combinatorial Structure ◽  
Geometrical Condition ◽  
The Third ◽  
Galois Plane ◽  
Hughes Plane

AbstractIn the first part of this paper, tests were described for determining which points of any line in a Galois plane II of order q2 are to be transferred to the conjugate line in order to transmute II into the corresponding Hughes plane Ω. In this part of the paper the tests are refined to provide, in relation to some fixed point in the central subplane δ0 of Ω (i) a simple geometrical condition of transfer for a certain set of ½q(q2−1) points of II and (ii) a simple aglebraic condition for the remaining points of II – δ0. These tests eliminate from the computation (for a given value of q) the necessity of calculating the third coordinates of ½q2 (q2−1) points in order to determine which are not-squares.

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