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Article

Quasi-Configurations Derived by Special Arrangements of Lines

by
Stefano Innamorati
Department of Industrial and Information Engineering and Economics, University of L’Aquila, Piazzale Ernesto Pontieri, 1, I-67100 L’Aquila, Italy
Axioms 2024, 13(5), 321; https://doi.org/10.3390/axioms13050321
Submission received: 2 April 2024 / Revised: 6 May 2024 / Accepted: 8 May 2024 / Published: 11 May 2024
(This article belongs to the Special Issue Theory of Curves and Knots with Applications)

Abstract

:
A quasi-configuration is a point–line incidence structure in which each point is incident with at least three lines and each line is incident with at least three points. We investigate derived quasi-configurations that arise both by duality and intersecting lines of three special arrangements of lines. Sets with few intersection numbers are provided.

1. Introduction

In finite geometry, the concepts of points and lines are translated from the continuous to the discrete world. A natural way to make this explicit is by starting with a geometry over the complex field ℂ and replacing the field ℂ by a finite field, 𝔽q, of order q, where q is a power of a prime number, thus obtaining a discrete object. The axiomatic approach to geometry contributes to this point of view, allowing geometries consisting only of a finite number of objects that have many applications in coding theory, data storage, or cryptography. Recall that a point–line incidence structure is a triple (P,L,I), where P is a set of points and L is a set of lines, together with a point–line incidence relation IP × L, where two points of P can be incident with at most one line of L, and two lines of L can be incident with at most one point of P. Throughout the paper, we only consider connected incidence structures, where any two elements of PL are connected via a path of incident elements.
One of the main problems in the theory of point–line incidence structures is to clarify the existence of regular ones, i.e., in which any line contains the same number of points, and any point is contained in the same number of lines. A (vr,bk)-configuration is a point–line incidence structure (P,L) where the size of P is v, the size of L is b, each point of P is contained in r lines of L and each line of L contains k points of P. In such a configuration, we have that vr = bk. If the number of points is equal to the number of lines, we call it a symmetric (nk) configuration.
A point–line incidence structure (P,L,I) is said to be a quasi-configuration of type p r 1 1 p r 2 2 p r m ( P ) m ( P ) , n k 1 1 n k 2 2 n k m ( L ) m ( L ) if P is a disjoint union of subsets Pi of cardinality pi with i = 1, 2, …, m(P), and L is a disjoint union of subsets Lj of cardinality nj with j = 1, 2, …, m(L), such that each point in Pi is incident with ri lines, and each line in Lj is incident with kj points. Quasi-configurations were introduced in [1] and used as building blocks for larger point–line incidence structures, see also [2]. Note that a quasi-configuration with m(P) = m(L) = 1 is a configuration, and if all the parameters are the same for points and lines we say that it is a symmetric quasi-configuration of type n k 1 1 n k 2 2 n k m ( L ) m ( L ) . Let 𝔽 denote a field and consider the projective plane ℙ2 over 𝔽. Let d ≥ 3 be an integer. An arrangement of d lines A is a set of d lines {L1, L2, …, Ld} in ℙ2, such that i = 1 d L i = . For i ≥ 2, an i-point is a point in A which belongs to exact i lines in A. We denote the number of i-points by ti. In characteristic zero, apart from pencil of three or more lines, only three types of complex line arrangements with t2 = 0 are currently known: Ceva, Klein, and Wiman arrangements.

2. Materials and Methods

A finite projective plane is a symmetric configuration, such that two points are contained in exactly one line, two lines meet in exactly one point, and there are four points, three by three not collinear. Finite projective planes are constructed using homogeneous coordinates with entries from a finite field 𝔽q. There are also planes not derived from finite fields, but they are beyond the scope of this research. We will study quasi-configurations which arise both by duality and intersecting lines of these special arrangements of lines, by replacing the field of complex numbers by the smallest finite field 𝔽q containing them. The idea is that if an arrangement of lines is special, then its derived quasi-configuration is also special. A k-set K in a finite projective plane is a set of k points. A k-set K is said to be symmetric if the number vi(P) of i-secant lines, through a point P of K is the same for any point P. A k-set K is said to be a blocking set if K meets every line. It is minimal if it properly contains no blocking sets. A k-set K is said to be of type (m1, m2, …, ms) if any line has exactly m points in common with K with m∈{m1, m2, …, ms} and every value occurs. The integers m1, m2, …, ms are called intersection numbers. Sets with few intersection numbers define projective linear codes with few weights that are useful in authentication codes, secret sharing schemes, and data storage systems. The paper falls into three sections. In the first, we will prove that, in characteristic 2, the Ceva (3) configuration closes, from the incidence point of view, in a projective plane of order four, and this, to our knowledge, seems to be novel. In the second section, we will show that, in characteristic 7, the Klein quasi-configuration is the set of the internal points of a conic of PG (2,7). Unfortunately, this representation is well known, see [3] (p. 7511), [4] (p. 342), and [5] (p. 121), but we provide direct proof arising from the Singer construction of the quasi-configuration. In the third section, we will prove that, in PG (2,19), the Wiman configuration gives rise to a symmetric minimal blocking 45-set of type (1,3,4,5), which, to our knowledge, seems to be novel.

3. Ceva Arrangements of Lines

Ceva(n) arrangements of lines are defined by the set of zeros of linear factor of the polynomial (xnyn)(xnzn)(ynzn), see [6]. This polynomial splits over a field 𝔽 containing a primitive root ω of unity of degree n, into the linear factors xωky, xωkz, yωkz, k = 0, 1, …, n − 1. Since dual Ceva (1) is a set of three collinear points and dual Ceva (2) is a complete quadrangle, suppose that the characteristic of the field 𝔽 is not three, and consider Ceva (3). The arrangement consists of the nine lines: xy = 0, xωy = 0, xω2y = 0, xz = 0, xω2z = 0, xωz = 0, yz = 0, yωz = 0, yω2z = 0. Dually, we get a set of nine points:
A: = (0,1,−1), B: = (0,1,−ω), C: = (0,1,−ω2);
P: = (1,0,−1), Q: = (1,0,−ω2), R: = (1,0,−ω);
X: = (1,−1,0), Y: = (1,−ω,0), Z: = (1,−ω2,0).
A direct check shows that the nine points of the above point matrix are contained in the twelve lines, three by three, that are rows, columns, and determinantal products. Thus, we get the matrix of the twelve lines:
ABC: x = 0, PQR: y = 0, XYZ: z = 0;
APX: x + y + z = 0, BQY: x + ω2y + ωz = 0, CRZ: x + ωy + ω2z = 0;
AQZ: x + ωy + ωz = 0, BRX: x + y + ω2z = 0, CPY: x + ω2y + z = 0;
CQX: x + y + ωz = 0, ARY: x + ω2y + ω2z = 0, BPZ: x + ωy + z = 0;
i.e., the Hesse (94,123) configuration as in Figure 1.
Since the characteristic of the field 𝔽 is not three, then each triple of parallel lines forms a different triangle, i.e., each row of the above line matrix is a triangle whose vertices are written in the rows of the matrix below:
D: = (0,0,1), E: = (1,0,0), F: = (0,1,0);
G: = (1,ω,ω2), H: = (1,ω2,ω), I: = (1,1,1);
L: = (1,ω,1), M: = (1,1,ω), N: = (1,ω2,ω2);
U: = (1,ω,ω), T: = (1,ω2,1), S: = (1,1,ω2).
The points and their alignments are shown in Figure 2.
A direct check shows that these twelve points are contained, four by four, in the nine lines of the Ceva arrangement:
DGLU: xω2y = 0, DIMS: xy = 0, DHNT: xωy = 0;
EGMT: yω2z = 0, EHSL: yωz = 0, EINU: yz = 0;
FGNS: xz = 0, FHMU: xω2z = 0, FILT: xωz = 0.
The 12 intersection points of the parallel lines and their alignments are shown in Figure 3.
In a direct check, if the characteristic of the field 𝔽 is two, then we have that
XDIMS, YDGLU, ZDHNT,
AEINU, BEGMT, CEHSL,
PFILT, QFGNS, RFHMU,
and we get a symmetric (215) configuration, i.e., a projective plane of order four, as shown in Figure 4.
In a direct check, if the characteristic of the field 𝔽 is two, then the Ceva (3) configuration closes, from incidence point of view, in a projective plane of order four, and this, to our knowledge, seems to be novel.
If the characteristic of the field 𝔽 is not two, then we get either, without considering two lines, a symmetric 9 4 1 12 5 2 quasi-configuration or a 21-set of type (0,1,2,4,5).

4. Klein Arrangement of Lines

The Klein arrangement of lines naturally arises from the subgroup PSL (2,7), the finite simple group of order 168, which is the automorphism group of the Klein quartic curve x3y + y3z + z3y = 0 of ℙ2, see [7,8]. It contains 21 involutions, each leaving a line fixed. The arrangement of these 21 lines is called the Klein arrangement. Let 𝔽 be a field containing a root ω of x2 + x + 2. The Klein arrangement consists of the 21 lines:
x = 0, y + z = 0, ωx + yz = 0, ωxy + z = 0, −y + z = 0, x + ωyz = 0, ωxyz = 0,
x + y + ωz = 0, ωx + y + z = 0, −x + ωy + z = 0, −xy + ωz = 0, x + z = 0, −x + ωyz = 0, x + ωy + z = 0,
x + z = 0, xy + ωz = 0, x + y + ωz = 0, z = 0, −x + y = 0, x + y = 0, y = 0.
Dually, we get a set of 21 points:
P1: = (1,0,0), P2: = (0,1,1), P3: = (1,ω−1,−ω−1), P4: = (1,−ω−1,ω−1), P5: = (0,1,−1), P6: = (1,ω,−1), P7: = (1,−ω−1,−ω−1),
P8: = (1,−1,−ω), P9: = (1,ω−1,ω−1), P10: = (1,−ω,−1), P11: = (1,1,−ω), P12: = (1,0,1), P13: = (1,−ω,1), P14: = (1,ω,1),
P15: = (1,0,−1), P16: = (1,−1,ω), P17: = (1,1,ω), P18: = (0,0,1), P19: = (1,−1,0), P20: = (1,1,0), P21: = (0,1,0).
A direct check shows either that the 21 points of the above point matrix are contained four by four in the 21 lines,
P2P5P18P21 x = 0, P1P12P15P18 y = 0, P1P19P20P21 z = 0, P1P3P4P5 y + z = 0, P12P13P14P21 xz = 0,
P11P17P18P20 xy = 0, P6P10P15P21 x + z = 0, P1P2P7P9 yz = 0, P8P16P18P19 x + y = 0,
P3P10P11P12 x − (ω + 1)yz = 0, P4P14P15P16 x + (ω + 1)y + z = 0, P9P13P15P17 x − (ω + 1)y + z = 0,
P7P10P16P20 xy + (ω + 1)z = 0, P2P4P6P11 (ω + 1)xy + z = 0, P5P7P11P14 (ω + 1)xyz = 0,
P5P8 P9P10 (ω + 1)x + y + z = 0, P3P6P17P19 x + y + (ω + 1)z = 0, P2P3P13P16 (ω + 1)x + yz = 0,
P6P7P8P12 x + (ω + 1)yz = 0, P4P8P13P20 xy − (ω + 1)z = 0, P9P11P14P19 x + y − (ω + 1)z = 0,
or that the 21 points of the above point matrix are contained three by three in the 28 lines,
P1P6P13 y + ωz = 0, P1P10P14 yωz = 0, P6P14P18 ωxy = 0, P1P8P17 ωyz = 0, P2P8P14 (−ω + 1)x + yz = 0,
P3P14P20 xy + (ω − 1)z = 0, P3P8P15 x + (−ω + 1)y + z = 0, P10P13P18 ωx + y = 0, P3P9P18 xωy = 0,
P6P9P20 xy + (−ω + 1)z = 0, P7P13P19 x + y + (ω − 1)z = 0, P3P7P21 x + ωz = 0, P4P7P18 x + ωy = 0, P2P15P20 xy + z = 0,
P7P11P15 x + (ω − 1)y + z = 0, P1P11P16 ωy + z = 0, P5P11P13 (ω − 1)x + y + z = 0, P8P11P21 ωx + z = 0, P2P12P19 x + yz = 0,
P5P15P19 x + y + z = 0, P4P9P21 xωz = 0, P4P10P19 x + y + (−ω + 1)z = 0, P2P10P17 (ω − 1)x + yz = 0,
P9P12P16 x + (−ω + 1)yz = 0, P4P12P17 x + (ω − 1)yz = 0, P5P6P16 (ω − 1)xyz = 0, P5P12P20 xyz = 0,
P16P17P21 ωxz = 0.
Thus, we get a ((218), 28 3 1 21 4 2 ) quasi-configuration.
Since 7 is the smallest order, with characteristic different from 2, of a finite field, which contains a root of x2 + x + 2, such that ℙ2 contains at least 21 points, let us consider the field 𝔽7. In 𝔽7, 3 is a root of x2 + x + 2. In order to write the cyclic structure of ℙ2: = PG (2,7), let ω be a primitive element of 𝔽73 over 𝔽7 and let f x = a 0 + a 1 x + a 2 x 2 + x 3 be its minimal polynomial over 𝔽7. The companion matrix C f of f is given by C f = 0 1 0 0 0 1 a 0 a 1 a 2 and it induces a Singer cycle γ of PG(2,7), cf. [9]. Let us consider a primitive polynomial with minimal weight, i.e., the minimal number of non-zero coefficients, among all primitives of that degree over 𝔽7, f(x) = x3 + 3x + 2, cf. [10]. The companion matrix C(f) is 0 1 0 0 0 1 5 4 0 . Let us consider the point ω 0 = x 0 x 1 x 2 = 1 0 0 . We get:
ω 2 = C ( f ) 2 ω 0 = C ( f ) ω 1 = 0 1 0 0 0 1 5 4 0 0 0 1 = 0 1 0 , ω 3 = C ( f ) 3 ω 0 = C ( f ) ω 2 = 0 1 0 0 0 1 5 4 0 0 1 0 = 1 0 4 , ω 4 = C ( f ) 4 ω 0 = C ( f ) ω 3 = 0 1 0 0 0 1 5 4 0 1 0 4 = 0 4 5 = 4 0 1 3 = 0 1 3 , ω 5 = C ( f ) 4 ω 0 = C ( f ) ω 4 = 0 1 0 0 0 1 5 4 0 0 1 3 = 1 3 4 ,
by continuing in this way, we obtain the cyclic structure of PG (2,7) as shown in Table 1.
Let us denote the points represented by ωi simply by i. Thus, the Singer group is isomorphic to the additive group Z57, the integers modulo 57. The sets of 21 points are:
P1 = (1,0,0) = 0, P2 = (0,1,1) = 53, P3 = (1,5,2) = 18, P4 = (1,2,5) = 48, P5 = (0,1,6) = 37, P6 = (1,3,6) = 39,
P7 = (1,2,2) = 24, P8 = (1,6,4) = 38, P9 = (1,5,5) = 16, P10 = (1,4,6) = 50, P11 = (1,1,4) = 54, P12 = (1,0,1) = 13,
P13 = (1,4,1) = 30, P14 = (1,3,1) = 27, P15 = (1,0,6) = 43, P16 = (1,6,3) = 49, P17 = (1,1,3) = 25, P18 = (0,0,1) = 1,
P19 = (1,6,0) = 42, P20 = (1,1,0) = 35, P21 = (0,1,0) = 2,
Now, select any line, for example, we choose the line 0: = x1 = 0, which contains the 8-set of points written in Table 2.
The remaining lines of the plane are found by adding 1 to each point of the preceding line beginning with 0 and using addition modulo 57. For the convenience of the reader, we represent the projective plane of order 7 as a set of four orthogonal arrays of the affine plane of order 7, with the intersection point of the elements of each parallel class indicated to the right of the row array and at the bottom of the column array.
Let us color the points of the 21-set green and the others red.
{0,1,2,13,16,18,24,25,27,30,35,37,38,39,42,43,48,49,50,53,54},
{3,4,5,6,7,8,9,10,11,12,14,15,17,19,20,21,22,23,26,28,29,31,32,33,34,36,40,41,44,45,46,47,51,52,55,56}.
The alignments of the Singer representation, written in Table 3, show that the Klein ( 21 7 1 , 28 3 1 21 4 2 ) quasi-configuration embedded in PG (2,7) is a symmetric 21-set of type (0,3,4), which are, by the result in [11], the internal points of a conic, see also [12,13,14].

5. Wiman Arrangement of Lines

Let 𝔽 be a field containing a root of x4x2 + 4 and sufficiently large such that the resulting 201 points are different. The Wiman arrangement of lines consists of 45 lines of ℙ2, with 36 quintuple points, 45 quadruple points, and 120 triple points, see [3,15,16] for a detailed description of the group action giving rise to it.
Since 19 is the smallest order, with characteristic different from 2, of a finite field, such that ℙ2 contains at least 201 points, let us consider the field 𝔽19. In 𝔽19, 3 is a root of x4x2 + 4. The dual of the Wiman arrangement of lines consists of the 45 points, see [16]:
P1: = (0,1,0), P2: = (1,16,15), P3: = (0,0,1), P4: = (1,3,15), P5: = (1,14,6), P6: = (1,5,6), P7: = (1,9,5), P8: = (1,10,5),
P9: = (1,3,4), P10: = (1,14,13), P11: = (1,5,4), P12: = (1,12,18), P13: = (1,9,14), P14: = (1,15,2), P15: = (1,15,17),
P16: = (1,16,4), P17: = (1,5,13), P18: = (1,14,4), P19: = (1,7,18), P20: = (1,10,14), P21: = (1,4,2), P22: = (1,4,17),
P23: = (1,1,12), P24: = (1,14,15), P25: = (1,5,15), P26: = (1,12,1), P27: = (1,11,0), P28: = (1,8,8), P29: = (1,7,1),
P30: = (1,10,17), P31: = (1,18,12), P32: = (1,8,0), P33: = (1,11,8), P34: = (1,9,17), P35: = (1,0,7), P36: = (1,0,0),
P37: = (1,10,2), P38: = (0,1,11), P39: = (1,8,11), P40: = (1,11,11), P41: = (1,9,2), P42: = (0,1,8), P43: = (1,18,7),
P44: = (1,0,12), P45: = (1,1,7).
In order to write the cyclic structure of PG (2,19), let ω be a primitive element of GF (193) over GF (19) and let f x = a 0 + a 1 x + a 2 x 2 + x 3 be its minimal polynomial over GF (19). The companion matrix C f of f is given by
C f = 0 1 0 0 0 1 a 0 a 1 a 2
and it induces a Singer cycle γ of PG (2,19), cf. [9]. Let us consider a primitive polynomial with minimal weight, i.e., the minimal number of non-zero coefficients, among all primitives of that degree over GF (19), f(x) = 17 + 15x2 + x3, cf. [10]. The companion matrix C(f) is 0 1 0 0 0 1 2 0 4 .
Let us consider the point ω 0 = x 0 x 1 x 2 = 1 0 0 . We get:
ω 1 = ω 0 C ( f ) = 1 0 0 0 1 0 0 0 1 2 0 4 = 0 1 0 . ω 2 = ω 0 C ( f ) 2 = ω 1 C ( f ) = 0 1 0 0 1 0 0 0 1 2 0 4 = 0 0 1 . ω 3 = ω 0 C ( f ) 3 = ω 2 C ( f ) = 0 0 1 0 1 0 0 0 1 2 0 4 = 2 0 4 = 2 1 0 2 = 1 0 2 ,
by continuing in this way, we obtain the cyclic structure of PG (2,19) as shown in Table 4.
Let us denote the points represented by ωi simply by i. Thus, the Singer group is isomorphic to the additive group Z381, the integers modulo 381.
P1: = (0,1,0) = 1, P2: = (1,16,15) = 35, P3: = (0,0,1) = 2, P4: = (1,3,15) = 19, P5: = (1,14,6) = 266, P6: = (1,5,6) = 343,
P7: = (1,9,5) = 311, P8: = (1,10,5) = 33, P9: = (1,3,4) = 232, P10: = (1,14,13) = 207, P11: = (1,5,4) = 237,
P12: = (1,12,18) = 199, P13: = (1,9,14) = 287, P14: = (1,15,2) = 254, P15: = (1,15,17) = 325, P16: = (1,16,4) = 346,
P17: = (1,5,13) = 271, P18: = (1,14,4) = 198, P19: = (1,7,18) = 41, P20: = (1,10,14) = 256, P21: = (1,4,2) = 159,
P22: = (1,4,17) = 279, P23: = (1,1,12) = 145, P24: = (1,14,15) = 28, P25: = (1,5,15) = 40, P26: = (1,12,1) = 209,
P27: = (1,11,0) = 272, P28: = (1,8,8) = 354, P29: = (1,7,1) = 148, P30: = (1,10,17) = 350,
P31: = (1,18,12) = 278, P32: = (1,8,0) = 267, P33: = (1,11,8) = 367, P34: = (1,9,17) = 363,
P35: = (1,0,7) = 253, P36: = (1,0,0) = 0, P37: = (1,10,2) = 128, P38: = (0,1,11) = 273,
P39: = (1,8,11) = 49, P40: = (1,11,11) = 7, P41: = (1,9,2) = 95, P42: = (0,1,8) = 268,
P43: = (1,18,7) = 352, P44: = (1,0,12) = 158, P45: = (1,1,7) = 324.
Now, select any line as the line at infinity, for example, we choose the line := x0 = 0. The remaining lines of the plane are found by adding 1 to each point of the preceding line beginning with as 0 and using addition modulo 381. Table 5 shows that the above 45 points are contained in 36 5-lines, 45 4-lines and 120 3-lines.
Thus, we get a ( 45 16 1 , 120 3 1 45 4 2 36 5 3 ) quasi-configuration embedded in PG (2,19), which is a symmetric minimal blocking 45-set of type (1,3,4,5), which, to our knowledge, seems to be novel.

6. Conclusions

Sets with few intersection numbers are connected with many theoretical and applied areas, such as coding theory, strongly regular graphs, association schemes, optimal multiple coverings, and secret sharing, cf. [17]. In this paper, sets with few intersection numbers are provided by quasi-configurations derived by special arrangements of lines.

Funding

This research received no external funding.

Data Availability Statement

The data presented in this study are available on request from the corresponding author.

Acknowledgments

The author acknowledge GNSAGA of INDAM for supporting research.

Conflicts of Interest

The author declares no conflicts of interest.

References

  1. Bokowski, J.; Pilaud, V. Quasi-configurations: Building blocks for point-line configurations. Ars Math. Contemp. 2016, 10, 99–112. [Google Scholar] [CrossRef]
  2. Berman, L.W.; Gévay, G.; Pisanski, T. On a new (21_4) polycyclic configuration. arXiv 2023, arXiv:2309.12992v2. [Google Scholar]
  3. Bauer, T.; Di Rocco, S.; Harbourne, B.; Huizenga, J.; Seceleanu, A.; Szemberg, T. Negative Curves on Symmetric Blowups of the Projective Plane, Resurgences, and Waldschmidt Constants. Int. Math. Res. Not. 2019, 2019, 7459–7514. [Google Scholar] [CrossRef]
  4. Grünbaum, B.; Rigby, J.F. The Real Configuration (214). J. Lond. Math. Soc. 1990, 2, 336–346. [Google Scholar] [CrossRef]
  5. Coxeter, H.S.M. My graph. Proc. London Math. Soc. 1983, 46, 117–136. [Google Scholar] [CrossRef]
  6. Ceva, G. De Lineis Rectis se Invicem Secantibus Statica Constructio, Mediolani, ex Typographia Ludouici Montiae, 1678. Available online: https://archive.org/details/ita-bnc-mag-00001346-001 (accessed on 1 April 2024).
  7. Klein, F. Uber die Transformation siebenter Ordnung der elliptischen Functionen. Math. Ann. 1879, 14, 428–471. [Google Scholar] [CrossRef]
  8. Gévay, G.; Pokora, P. Klein’s arrangements of lines and conics. Beitr Algebra Geom 2023, 65, 393–414. [Google Scholar] [CrossRef]
  9. Singer, J. A theorem in finite projective geometry and some applications to number theory. Trans. Amer. Math. Soc. 1938, 43, 377–385. Available online: https://www.ams.org/journals/tran/1938-043-03/S0002-9947-1938-1501951-4/S0002-9947-1938-1501951-4.pdf (accessed on 1 April 2024). [CrossRef]
  10. Hansen, T.; Mullen, G.L. Primitive Polynomials Over Finite Fields. Math. Comput. 1992, 59, 639–643. [Google Scholar] [CrossRef]
  11. De Clerck, F.; De Feyter, N. A characterization of the sets of internal and external points of a conic. Eur. J. Comb. 2007, 28, 1910–1921. [Google Scholar] [CrossRef]
  12. Hill, R.; Love, C.P. On the (22, 4)-arcs in PG(2, 7) and related codes. Discrete Math. 2003, 266, 253–261. [Google Scholar] [CrossRef]
  13. Van de Voorde, G. On sets without tangents and exterior sets of a conic. Discrete Math. 2011, 311, 2253–2258. [Google Scholar] [CrossRef]
  14. Bouyukliev, I.; Cheon, E.J.; Maruta, T.; Okazaki, T. On the (29, 5)-Arcs in PG(2,7) and Some Generalized Arcs in PG(2,q). Mathematics 2020, 8, 320. [Google Scholar] [CrossRef]
  15. Wiman, A. Zur Theorie der endlichen Gruppen von birationalen Transformationen in der Ebene. Math. Annalen 1896, 48, 195–240. [Google Scholar] [CrossRef]
  16. Bauer, T.; Di Rocco, S.; Harbourne, B.; Huizenga, J.; Lundman, A.; Pokora, P.; Szemberg, T. Bounded Negativity and Arrangements of Lines. Int. Math. Res. Not. 2015, 2015, 9456–9471. [Google Scholar] [CrossRef]
  17. Durante, N. On sets with few intersection numbers in finite projective and affine spaces. Electron. J. Combin. 2014, 21, P4–13. [Google Scholar] [CrossRef]
Figure 1. The Hesse (94,123) configuration.
Figure 1. The Hesse (94,123) configuration.
Axioms 13 00321 g001
Figure 2. The Hesse (94,123) configuration with the 12 intersection points of the parallel lines.
Figure 2. The Hesse (94,123) configuration with the 12 intersection points of the parallel lines.
Axioms 13 00321 g002
Figure 3. The 12 intersection points of the parallel lines are contained in the nine lines of the Ceva arrangement.
Figure 3. The 12 intersection points of the parallel lines are contained in the nine lines of the Ceva arrangement.
Axioms 13 00321 g003
Figure 4. The projective plane of order four.
Figure 4. The projective plane of order four.
Axioms 13 00321 g004
Table 1. The cyclic structure of the points of PG (2,7).
Table 1. The cyclic structure of the points of PG (2,7).
ω0 = (1,0,0)
ω1 = (0,0,1)ω2 = (0,1,0)ω3 = (1,0,4)ω4 = (0,1,3)ω5 = (1,3,4)ω6 = (1,6,1)ω7 = (1,6,6)ω8 = (1,1,6)
ω9 = (1,6,2)ω10 = (1,5,6)ω11 = (1,4,5)ω12 = (1,3,0)ω13 = (1,0,1)ω14 = (0,1,5)ω15 = (1,5,4)ω16 = (1,5,5)
ω17 = (1,1,5)ω18 = (1,5,2)ω19 = (1,6,5)ω20 = (1,2,6)ω21 = (1,3,3)ω22 = (1,1,1)ω23 = (1,1,2)ω24 = (1,2,2)
ω25 = (1,1,3)ω26 = (1,3,2)ω27 = (1,3,1)ω28 = (1,5,1)ω29 = (1,3,5)ω30 = (1,4,1)ω31 = (1,2,0)ω32 = (1,0,3)
ω33 = (0,1,4)ω34 = (1,4,4)ω35 = (1,1,0)ω36 = (1,0,2)ω37 = (0,1,6)ω38 = (1,6,4)ω39 = (1,3,6)ω40 = (1,2,1)
ω41 = (1,4,3)ω42 = (1,6,0)ω43 = (1,0,6)ω44 = (0,1,2)ω45 = (1,2,4)ω46 = (1,2,3)ω47 = (1,5,3)ω48 = (1,2,5)
ω49 = (1,6,3)ω50 = (1,4,6)ω51 = (1,5,0)ω52 = (1,0,5)ω53 = (0,1,1)ω54 = (1,1,4)ω55 = (1,4,2)ω56 = (1,4,0)
Table 2. The starting line.
Table 2. The starting line.
00131332364352
Table 3. The projective plane PG (2,7) in which the points of the 21-set are colored green and its complementary set colored red.
Table 3. The projective plane PG (2,7) in which the points of the 21-set are colored green and its complementary set colored red.
512212627293912515343845543
64273894919646553539164
835222517542322424847332650
185116281510472823729411431
37253414334427102537245619
4156341150553030531220174918
48312420464045401144951821
013
333545117182732484951423273443
341063117264425071020839
46212549472411124541264714
830291542442165309333125
1238503740231619352641155340
191451225552017283745215542
284853395695429444622563818
3652
Table 4. The cyclic structure of the points of PG (2,19).
Table 4. The cyclic structure of the points of PG (2,19).
ω0 = (1,0,0)ω1 = (0,1,0)ω2 = (0,0,1)
ω3 = (1,0,2)ω4 = (1,5,2)ω5 = (1,5,8)ω6 = (1,6,13)ω7 = (1,11,11)ω8 = (1,13,12)ω9 = (1,4,16)ω10 = (1,3,14)ω11 = (1,17,15)
ω12 = (1,7,7)ω13 = (1,15,12)ω14 = (1,4,5)ω15 = (1,2,10)ω16 = (1,1,4)ω17 = (1,12,14)ω18 = (1,17,16)ω19 = (1,3,15)ω20 = (1,7,4)
ω21 = (1,12,10)ω22 = (1,1,14)ω23 = (1,17,0)ω24 = (0,1,17)ω25 = (1,0,16)ω26 = (1,3,2)ω27 = (1,5,18)ω28 = (1,14,15)ω29 = (1,7,5)
ω30 = (1,2,16)ω31 = (1,3,8)ω32 = (1,6,1)ω33 = (1,10,5)ω34 = (1,2,3)ω35 = (1,16,15)ω36 = (1,7,0)ω37 = (0,1,7)ω38 = (1,0,17)
ω39 = (1,14,2)ω40 = (1,5,15)ω41 = (1,7,18)ω42 = (1,9,8)ω43 = (1,6,18)ω44 = (1,9,18)ω45 = (1,9,7)ω46 = (1,15,4)ω47 = (1,12,11)
ω48 = (1,13,6)ω49 = (1,8,11)ω50 = (1,13,11)ω51 = (1,13,0)ω52 = (0,1,13)ω53 = (1,0,13)ω54 = (1,11,2)ω55 = (1,5,0)ω56 = (0,1,5)
ω57 = (1,0,4)ω58 = (1,12,2)ω59 = (1,5,5)ω60 = (1,2,12)ω61 = (1,4,10)ω62 = (1,1,6)ω63 = (1,8,10)ω64 = (1,1,10)ω65 = (1,1,3)
ω66 = (1,16,18)ω67 = (1,9,13)ω68 = (1,11,6)ω69 = (1,8,14)ω70 = (1,17,5)ω71 = (1,2,17)ω72 = (1,14,11)ω73 = (1,13,13)ω74 = (1,11,12)
ω75 = (1,4,8)ω76 = (1,6,7)ω77 = (1,15,16)ω78 = (1,3,9)ω79 = (1,18,18)ω80 = (1,9,12)ω81 = (1,4,0)ω82 = (0,1,4)ω83 = (1,0,14)
ω84 = (1,17,2)ω85 = (1,15,11)ω86 = (1,13,10)ω87 = (1,1,15)ω88 = (1,7,9)ω89 = (1,18,14)ω90 = (1,17,4)ω91 = (1,12,16)ω92 = (1,3,0)
ω93 = (0,1,3)ω94 = (1,0,18)ω95 = (1,9,2)ω96 = (1,5,9)ω97 = (1,18,16)ω98 = (1,3,18)ω99 = (1,9,10)ω100 = (1,1,11)ω101 = (1,13,15)
ω102 = (1,7,17)ω103 = (1,14,5)ω104 = (1,2,11)ω105 = (1,13,11)ω106 = (1,18,8)ω107 = (1,6,15)ω108 = (1,7,6)ω109 = (1,8,1)ω110 = (1,10,6)
ω111 = (1,8,6)ω112 = (1,8,9)ω113 = (1,18,14)ω114 = (1,11,10)ω115 = (1,1,13)ω116 = (1,11,13)ω117 = (1,11,9)ω118 = (1,18,10)ω119 = (1,1,1)
ω120 = (1,10,12)ω121 = (1,4,4)ω122 = (1,12,12)ω123 = (1,4,12)ω124 = (1,4,18)ω125 = (1,9,0)ω126 = (0,1,9)ω127 = (1,0,1)ω128 = (1,10,2)
ω129 = (1,5,14)ω130 = (1,17,11)ω131 = (1,13,14)ω132 = (1,17,14)ω133 = (1,17,6)ω134 = (1,8,5)ω135 = (1,2,18)ω136 = (1,9,1)ω137 = (1,10,16)
ω138 = (1,3,13)ω139 = (1,11,16)ω140 = (1,3,16)ω141 = (1,3,11)ω142 = (1,13,3)ω143 = (1,16,1)ω144 = (1,10,10)ω145 = (1,1,12)ω146 = (1,4,6)
ω147 = (1,8,15)ω148 = (1,7,1)ω149 = (1,10,15)ω150 = (1,7,15)ω151 = (1,7,13)ω152 = (1,11,3)ω153 = (1,16,7)ω154 = (1,15,14)ω155 = (1,17,10)
ω156 = (1,1,0)ω157 = (0,1,1)ω158 = (1,0,12)ω159 = (1,4,2)ω160 = (1,5,3)ω161 = (1,16,6)ω162 = (1,8,16)ω163 = (1,3,7)ω164 = (1,15,9)
ω165 = (1,18,6)ω166 = (1,8,13)ω167 = (1,11,14)ω168 = (1,17,18)ω169 = (1,9,3)ω170 = (1,16,13)ω171 = (1,11,7)ω172 = (1,15,15)ω173 = (1,7,12)
ω174 = (1,4,11)ω175 = (1,13,16)ω176 = (1,3,3)ω177 = (1,16,12)ω178 = (1,4,9)ω179 = (1,18,17)ω180 = (1,14,17)ω181 = (1,15,3)ω182 = (1,16,14)
ω183 = (1,17,8)ω184 = (1,6,9)ω185 = (1,18,15)ω186 = (1,7,14)ω187 = (1,17,7)ω188 = (1,15,10)ω189 = (1,1,17)ω190 = (1,14,16)ω191 = (1,3,6)
ω192 = (1,8,7)ω193 = (1,15,8)ω194 = (1,6,16)ω195 = (1,3,1)ω196 = (1,10,13)ω197 = (1,11,17)ω198 = (1,14,4)ω199 = (1,12,18)ω200 = (1,9,15)
ω201 = (1,7,8)ω202 = (1,6,6)ω203 = (1,8,12)ω204 = (1,4,15)ω205 = (1,7,11)ω206 = (1,13,17)ω207 = (1,14,13)ω208 = (1,11,4)ω209 = (1,12,1)
ω210 = (1,10,8)ω211 = (1,6,5)ω212 = (1,2,14)ω213 = (1,17,17)ω214 = (1,14,12)ω215 = (1,4,1)ω216 = (1,10,4)ω217 = (1,12,8)ω218 = (1,6,17)
ω219 = (1,14,10)ω220 = (1,1,16)ω221 = (1,3,5)ω222 = (1,2,8)ω223 = (1,6,14)ω224 = (1,17,9)ω225 = (1,18,4)ω226 = (1,12,9)ω227 = (1,18,9)
ω228 = (1,18,3)ω229 = (1,16,5)ω230 = (1,2,15)ω231 = (1,7,16)ω232 = (1,3,4)ω233 = (1,12,0)ω234 = (0,1,12)ω235 = (1,0,6)ω236 = (1,8,2)
ω237 = (1,5,4)ω238 = (1,12,5)ω239 = (1,2,7)ω240 = (1,15,13)ω241 = (1,11,15)ω242 = (1,7,3)ω243 = (1,16,0)ω244 = (0,1,16)ω245 = (1,0,5)
ω246 = (1,2,2)ω247 = (1,5,12)ω248 = (1,4,3)ω249 = (1,16,9)ω250 = (1,18,5)ω251 = (1,2,0)ω252 = (0,1,2)ω253 = (1,0,7)ω254 = (1,15,2)
ω255 = (1,5,1)ω256 = (1,10,14)ω257 = (1,17,1)ω258 = (1,10,1)ω259 = (1,10,7)ω260 = (1,15,0)ω261 = (0,1,15)ω262 = (1,0,9)ω263 = (1,18,2)
ω264 = (1,5,16)ω265 = (1,3,17)ω266 = (1,14,6)ω267 = (1,8,0)ω268 = (0,1,8)ω269 = (1,0,8)ω270 = (1,6,2)ω271 = (1,5,13)ω272 = (1,11,0)
ω273 = (0,1,11)ω274 = (1,0,15)ω275 = (1,7,2)ω276 = (1,5,18)ω277 = (1,9,9)ω278 = (1,18,12)ω279 = (1,4,17)ω280 = (1,14,1)ω281 = (1,10,9)
ω282 = (1,18,11)ω283 = (1,13,8)ω284 = (1,6,4)ω285 = (1,12,17)ω286 = (1,14,18)ω287 = (1,9,14)ω288 = (1,17,3)ω289 = (1,16,8)ω290 = (1,6,3)
ω291 = (1,16,3)ω292 = (1,16,11)ω293 = (1,13,1)ω294 = (1,10,18)ω295 = (1,9,16)ω296 = (1,3,10)ω297 = (1,1,5)ω298 = (1,2,4)ω299 = (1,12,7)
ω300 = (1,15,11)ω301 = (1,13,7)ω302 = (1,15,7)ω303 = (1,15,18)ω304 = (1,9,4)ω305 = (1,12,15)ω306 = (1,7,10)ω307 = (1,1,9)ω308 = (1,18,1)
ω309 = (1,10,11)ω310 = (1,13,18)ω311 = (1,9,5)ω312 = (1,2,1)ω313 = (1,10,3)ω314 = (1,16,10)ω315 = (1,1,18)ω316 = (1,9,11)ω317 = (1,13,5)
ω318 = (1,2,9)ω319 = (1,18,0)ω320 = (0,1,18)ω321 = (1,0,11)ω322 = (1,13,2)ω323 = (1,5,10)ω324 = (1,1,7)ω325 = (1,15,17)ω326 = (1,14,3)
ω327 = (1,16,17)ω328 = (1,14,17)ω329 = (1,14,8)ω330 = (1,6,10)ω331 = (1,1,8)ω332 = (1,6,8)ω333 = (1,6,0)ω334 = (0,1,6)ω335 = (1,0,10)
ω336 = (1,1,2)ω337 = (1,5,7)ω338 = (1,15,1)ω339 = (1,10,0)ω340 = (0,1,10)ω341 = (1,0,3)ω342 = (1,16,2)ω343 = (1,5,6)ω344 = (1,8,4)
ω345 = (1,12,3)ω346 = (1,16,4)ω347 = (1,12,4)ω348 = (1,12,13)ω349 = (1,11,1)ω350 = (1,10,17)ω351 = (1,14,9)ω352 = (1,18,7)ω353 = (1,15,6)
ω354 = (1,8,8)ω355 = (1,6,12)ω356 = (1,4,7)ω357 = (1,15,5)ω358 = (1,2,13)ω359 = (1,11,5)ω360 = (1,2,5)ω361 = (1,2,6)ω362 = (1,8,18)
ω363 = (1,9,17)ω364 = (1,14,14)ω365 = (1,17,12)ω366 = (1,4,13)ω367 = (1,11,8)ω368 = (1,6,11)ω369 = (1,13,4)ω370 = (1,12,6)ω371 = (1,8,3)
ω372 = (1,16,16)ω373 = (1,3,12)ω374 = (1,4,14)ω375 = (1,17,3)ω376 = (1,11,18)ω377 = (1,9,6)ω378 = (1,8,17)ω379 = (1,14,0)ω380 = (0,1,14)
Table 5. The cyclic structure of the lines of PG (2,19) in which the points of the 45-set are colored red.
Table 5. The cyclic structure of the lines of PG (2,19) in which the points of the 45-set are colored red.
012243752568293126157234244252261268273320334340380
1023253853578394127158235245253262269274321335341
2134263954588495128159236246254263270275322336342
3245274055598596129160237247255264271276323337343
4356284156608697130161238248256265272277324338344
5467294257618798131162239249257266273278325339345
6578304358628899132163240250258267274279326340346
76893144596389100133164241251259268275280327341347
879103245606490101134165242252260269276281328342348
9810113346616591102135166243253261270277282329343349
10911123447626692103136167244254262271278283330344350
111012133548636793104137168245255263272279284331345351
121113143649646894105138169246256264273280285332346352
131214153750656995106139170247257265274281286333347353
141315163851667096107140171248258266275282287334348354
151416173952677197108141172249259267276283288335349355
161517184053687298109142173250260268277284289336350356
171618194154697399110143174251261269278285290337351357
1817192042557074100111144175252262270279286291338352358
1918202143567175101112145176253263271280287292339353359
2019212244577276102113146177254264272281288293340354360
2120222345587377103114147178255265273282289294341355361
2221232446597478104115148179256266274283290295342356362
2322242547607579105116149180257267275284291296343357363
2423252648617680106117150181258268276285292297344358364
2524262749627781107118151182259269277286293298345359365
2625272850637882108119152183260270278287294299346360366
2726282951647983109120153184261271279288295300347361367
2827293052658084110121154185262272280289296301348362368
2928303153668185111122155186263273281290297302349363369
3029313254678286112123156187264274282291298303350364370
3130323355688387113124157188265275283292299304351365371
3231333456698488114125158189266276284293300305352366372
3332343557708589115126159190267277285294301306353367373
3433353658718690116127160191268278286295302307354368374
3534363759728791117128161192269279287296303308355369375
3635373860738892118129162193270280288297304309356370376
3736383961748993119130163194271281289298305310357371377
3837394062759094120131164195272282290299306311358372378
3938404163769195121132165196273283291300307312359373379
4039414264779296122133166197274284292301308313360374380
41040424365789397123134167198275285293302309314361375
42141434466799498124135168199276286294303310315362376
43242444567809599125136169200277287295304311316363377
443434546688196100126137170201278288296305312317364378
454444647698297101127138171202279289297306313318365379
465454748708398102128139172203280290298307314319366380
4706464849718499103129140173204281291299308315320367
48174749507285100104130141174205282292300309316321368
49284850517386101105131142175206283293301310317322369
50394951527487102106132143176207284294302311318323370
514105052537588103107133144177208285295303312319324371
525115153547689104108134145178209286296304313320325372
536125254557790105109135146179210287297305314321326373
547135355567891106110136147180211288298306315322327374
558145456577992107111137148181212289299307316323328375
569155557588093108112138149182213290300308317324329376
5710165658598194109113139150183214291301309318325330377
5811175759608295110114140151184215292302310319326331378
5912185860618396111115141152185216293303311320327332379
6013195961628497112116142153186217294304312321328333380
61014206062638598113117143154187218295305313322329334
62115216163648699114118144155188219296306314323330335
632162262646587100115119145156189220297307315324331336
643172363656688101116120146157190221298308316325332337
654182464666789102117121147158191222299309317326333338
665192565676890103118122148159192223300310318327334339
676202666686991104119123149160193224301311319328335340
687212767697092105120124150161194225302312320329336341
698222868707193106121125151162195226303313321330337342
709232969717294107122126152163196227304314322331338343
7110243070727395108123127153164197228305315323332339344
7211253171737496109124128154165198229306316324333340345
7312263272747597110125129155166199230307317325334341346
7413273373757698111126130156167200231308318326335342347
7514283474767799112127131157168201232309319327336343348
76152935757778100113128132158169202233310320328337344349
77163036767879101114129133159170203234311321329338345350
78173137777980102115130134160171204235312322330339346351
79183238788081103116131135161172205236313323331340347352
80193339798182104117132136162173206237314324332341348353
81203440808283105118133137163174207238315325333342349354
82213541818384106119134138164175208239316326334343350355
83223642828485107120135139165176209240317327335344351356
84233743838586108121136140166177210241318328336345352357
85243844848687109122137141167178211242319329337346353358
86253945858788110123138142168179212243320330338347354359
87264046868889111124139143169180213244321331339348355360
88274147878990112125140144170181214245322332340349356361
89284248889091113126141145171182215246323333341350357362
90294349899192114127142146172183216247324334342351358363
91304450909293115128143147173184217248325335343352359364
92314551919394116129144148174185218249326336344353360365
93324652929495117130145149175186219250327337345354361366
94334753939596118131146150176187220251328338346355362367
95344854949697119132147151177188221252329339347356363368
96354955959798120133148152178189222253330340348357364369
97365056969899121134149153179190223254331341349358365370
983751579799100122135150154180191224255332342350359366371
9938525898100101123136151155181192225256333343351360367372
10039535999101102124137152156182193226257334344352361368373
101405460100102103125138153157183194227258335345353362369374
102415561101103104126139154158184195228259336346354363370375
103425662102104105127140155159185196229260337347355364371376
104435763103105106128141156160186197230261338348356365372377
105445864104106107129142157161187198231262339349357366373378
106455965105107108130143158162188199232263340350358367374379
107466066106108109131144159163189200233264341351359368375380
1080476167107109110132145160164190201234265342352360369376
1091486268108110111133146161165191202235266343353361370377
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MDPI and ACS Style

Innamorati, S. Quasi-Configurations Derived by Special Arrangements of Lines. Axioms 2024, 13, 321. https://doi.org/10.3390/axioms13050321

AMA Style

Innamorati S. Quasi-Configurations Derived by Special Arrangements of Lines. Axioms. 2024; 13(5):321. https://doi.org/10.3390/axioms13050321

Chicago/Turabian Style

Innamorati, Stefano. 2024. "Quasi-Configurations Derived by Special Arrangements of Lines" Axioms 13, no. 5: 321. https://doi.org/10.3390/axioms13050321

APA Style

Innamorati, S. (2024). Quasi-Configurations Derived by Special Arrangements of Lines. Axioms, 13(5), 321. https://doi.org/10.3390/axioms13050321

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