Partitioning AG(2,q), q≡7 Mod 12, into MöbiusKantor Configurations and One Point

Abstract

By the cyclic structure of the affine plane AG(2,q), q≡7 mod 12, a mixed partition into a set of Möbius–Kantor configurations and a one-point set is provided. This generalizes a 2006 result of L. Berardi and T. Masini, who partitioned the affine plane of order 7 into a set of Möbius–Kantor configurations and a one-point set.

1. Introduction

Finite geometry is a fundamental concept of combinatorics, playing a crucial role in several mathematical contexts and having numerous applications in various fields, including coding theory, computer science and cryptography, particularly in the design of cryptographic protocols and algorithms. A problem in finite geometry that has been widely investigated is the possibility of partitioning the affine or projective space into objects that have the same geometric properties cf. [1,2,3,4,5,6,7]. For arithmetical reasons, sometimes it is not possible to partition a given geometry uniformly, although a uniform partition may be possible if one or more objects of a given kind are removed. We shall call such a non-uniform partition a mixed partition of a geometry, cf. [8,9,10,11,12,13]. Let p be prime and q = p^h, h ≥ 1. Then, ${\mathbb{F}}_q$ denotes the finite field of order q. A vector plane over ${\mathbb{F}}_q$ will be denoted as V(2,q). The affine plane over the field ${\mathbb{F}}_q$, denoted as AG(2,${\mathbb{F}}_q$), shortly, AG(2,q), is an incidence geometry of which the elements are the additive cosets of the vector subspaces of V(2,q). The incidence is symmetrized set-theoretic containment. As such, a point of AG(2,q) is represented by a unique vector of V(2,q), and a line is represented by a unique coset of a 1-dimensional vector subspace. Finite affine planes cannot be cyclic because they do not have the same number of lines and points. However, if we choose a single point (say ∞) from an affine plane, together with all the lines through that point, we can define a collineation σ such that the cyclic group generated by σ fixes ∞ and acts regularly on the set of the remaining points and on the set of remaining lines, cf. [14,15,16]. A Möbius–Kantor configuration 8₃ is a configuration consisting of eight points and eight lines with three points on each line and three lines on each point. It is not possible to draw points and lines having this pattern of incidences in the Euclidean plane. But it is possible in the complex plane. In AG(2,q), it is well-known that a Möbius–Kantor configuration 8₃ occurs for q≡0, 1 (mod 3), cf. [17]. Regarding 4, the smallest order q with q≡1 mod 3, in [18], the authors proved that AG(2,4) contains a partition into two Möbius–Kantor configurations 8₃. Taking into account 7, the second order q with q≡1 mod 3, in a 2006 paper, L. Berardi and T. Masini, cf. [19], showed by incidence properties that the affine plane AG(2,7) allows a mixed partition into one point ∞ and 6 Möbius–Kantor configurations 8₃. In this paper, through the cyclic structure of the affine plane AG(2,q), q≡7 mod 12, a mixed partition into a set of Möbius–Kantor configurations and a one-point set is provided. As usual, here a quadrilateral consists of four points in general positions, i.e., no three collinear. The remainder of this paper is structured as follows. Section 2 introduces the methodology used for the cyclic representation of the affine plane AG(2,q). Section 3 presents our findings, synthesizing key insights. Section 4 shows the construction in AG(2,7) with relevant figures and tables. Section 5, by our findings, derives a mixed partition in AG(2,19). In Section 6, we draw a discussion and put forward future research.

2. The Cyclic Representation of AG(2,q)

In this section, we describe the cyclic structure of the affine plane of order q. Let us consider a primitive polynomial of degree two in ${\mathbb{F}}_q$, p(x) = ax² + bx + c, q odd. The elements ${\displaystyle \frac{-b\pm \sqrt{b^2-4ac}}{2a}}$ are roots of p(x), and then they are primitive elements of ${\mathbb{F}}_{q^2}$. Since the element ${s=b}^2-4ac$ is a non-square element of ${\mathbb{F}}_q$, we can identify any point (x,y) of AG(2,q) with the element z = x + yr of with r² = s in the quadratic extension ${\mathbb{F}}_{q^2}$ of ${\mathbb{F}}_q$. Let α = α₁ + α₂r denote a primitive element of ${\mathbb{F}}_{q^2}$. The map σ: z–>αz is the linear collineation $\left(\begin{array}{c}x\prime \\ y\prime \end{array}\right)=\left(\begin{array}{cc}{\alpha }_1& {\alpha }_{2}s\\ {\alpha }_2& {\alpha }_1\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)$ of AG(2,q) because

z’ = x’ + y’r = σ(z) = αz = (α₁ + α₂r)·(x + yr) = (α₁x + α₂ys) + (α₁y + α₂x)r.

The cyclic group $〈\sigma 〉,$ generated by σ, has order q² − 1, fixes the point $\left(0,0\right)$, and $〈\sigma 〉$ is sharply transitive on the points of AG(2,q) distinct from $\left(0,0\right)$. If we choose the point $\left(1,0\right)$ as base point, then any point $\left(x,y\right)\ne \left(0,0\right)$ can be obtained uniquely as $\left(x,y\right)={\sigma }^i\left(1,0\right)$, with ${i\in \mathbb{Z}}_{q^2-1}$. Since the points of AG(2,q) can be represented by the set $\left\{\left(0,0\right)\right\}\cup \Omega =\left\{\left(0,0\right)\right\}\cup \left\{{\sigma }^i\left(1,0\right)| 0\le i\le q^2-2\right\}$, any point ${\sigma }^i\left(1,0\right)$can be identified with the exponent  $i, 0\le i\le q^2-2,$ and so with the elements of the additive cyclic group ${\mathbb{Z}}_{q^2-1}$. Therefore, the set of points of AG(2,q) is $\left\{\left(0,0\right)\right\}\cup \left\{0,1,\dots ,q^2-2\right\}.$

The $q^2-1$ lines not containing the point $\left(0,0\right)$ of the plane are found by selecting any line; for example, by choosing the line ${\mathcal{l}}_0≔x=1$, and by adding 1 to each point of the preceding line, beginning with $\mathcal{l}$₀ and using addition modulo $q^2-1$. Any line ℓ not containing the point $\left(0,0\right)$ is an affine difference set $\mathcal{l}=\left\{d_1,\dots ,d_q\right\}$ of ${\mathbb{Z}}_{q^2-1}$ relative to the subgroup ${\mathbb{Z}}_{q-1}$ of ${\mathbb{Z}}_{q^2-1}$, i.e., the set of differences $D=\left\{d_h-d_k :d_h,d_k\in \mathcal{l},h\ne k\right\}$ contains each element of ${\mathbb{Z}}_{q^2-1}-{\mathbb{Z}}_{q-1}$ exactly once but no element of ${\mathbb{Z}}_{q-1}$. The remaining q+1 lines of AG(2,q) containing the point $\left(0,0\right)$ are obtained tacking as base line, $m_0$, the union of $\left\{\left(0,0\right)\right\}$ with the unique subgroup of order q−1, ${\mathbb{Z}}_{q-1}$, of ${\mathbb{Z}}_{q^2-1}$. The remaining q lines containing the point $\left(0,0\right)$ of the plane are found by adding 1 to each point, different from $\left(0,0\right)$, of the preceding line, beginning with $m_0$, and using addition modulo$q^2-1$. Therefore, the lines of AG(2,q) are of two types, according to whether the line contains the point $\left(0,0\right)$ or not:

• The $i\mathrm{t}\mathrm{h}$ line ${\mathcal{l}}_i$, noncontaining the point $\left(0,0\right)$ with $i\in \left\{0,1,\dots ,q^2-2\right\}$, is the set

  $${\mathcal{l}}_i=\left\{d_1+i,\dots ,d_q+i\right\} \mathrm{m}\mathrm{o}\mathrm{d} q^2-1$$

  where $d_h$, $h\in \left\{1,2,\dots ,q\right\}$, are q integers modulo $q^2-1$, with $d_h\not\equiv 0 \mathrm{m}\mathrm{o}\mathrm{d} q+1$, that belong to an affine difference set of order q, i.e., the $q^2-q$ differences $d_h-d_k \mathrm{m}\mathrm{o}\mathrm{d} q^2-1$, $h,k\in \left\{1,2,\dots ,q\right\}, h\ne k$, are such that $d_h-d_k\not\equiv 0 \mathrm{m}\mathrm{o}\mathrm{d} q+1$.
• The $j\mathrm{t}\mathrm{h}$ line $m_j$, containing the point $\left(0,0\right)$ with $j\in \left\{0,1,\dots ,q\right\}$, is the set

  $$m_j=\left\{\left(0,0\right)\right\}\cup \left\{j,\left(q+1\right)+j,2\left(q+1\right)+j,\dots ,\left(q-2\right)\left(q+1\right)+j\right\} \mathrm{m}\mathrm{o}\mathrm{d} q^2-1,$$

  i.e., the other q−1 points, different from $\left(0,0\right)$, are the points $i, i\in \left\{0,1,\dots ,q^2-2\right\}$, with $i\equiv j \mathrm{m}\mathrm{o}\mathrm{d} q+1$.

3. The Construction of the Mixed Partition of AG(2,q), q≡7 Mod 12

In this section, we partition the affine plane of order q into a set of Möbius–Kantor configurations and a one-point set. We prove the following

Theorem 1.

Any affine plane AG(2,q) with q≡7 mod 12, admits a partition into a set of Möbius- Kantor configurations 8₃ plus a set consisting only of one point.

Proof. 

Let us suppose that q≡7 mod 12. The ${\displaystyle \frac{q^2-1}{4}}$ cosets of the unique subgroup of order 4, ${\mathbb{Z}}_4$, of ${\mathbb{Z}}_{q^2-1}$, define a partition $\left\{C_0,C_1,\dots ,C_{{\displaystyle \frac{q^2-5}{4}}}\right\}$ of AG(2,q)−$\left\{\left(0,0\right)\right\}$ in ${\displaystyle \frac{q^2-1}{4}}$ quadrilaterals where $C_0=\left\{0,{\displaystyle \frac{q^2-1}{4}},{\displaystyle \frac{q^2-1}{2}},{\displaystyle \frac{3\left(q^2-1\right)}{4}}\right\}$. We find that there is at least one quadrilateral

$$\begin{gathered} C_{2j+1}={\sigma }^{2j+1}\left(C_0\right)=\left\{2j+1,{\displaystyle \frac{q^2-1}{4}}+2j+1,{\displaystyle \frac{q^2-1}{2}}+2j+1,{\displaystyle \frac{3\left(q^2-1\right)}{4}}+2j+1\right\} \\ =\left\{2j+1,{\displaystyle \frac{q^2+8j+3}{4}},{\displaystyle \frac{q^2+4j+1}{2}},{\displaystyle \frac{3q^2+8j+1}{4}}\right\}, j\in \left\{0,1,\dots ,{\displaystyle \frac{q^2-9}{8}}\right\} \end{gathered}$$

which is mutually inscribed and circumscribed with $C_0$. Thus, the union $C_0\cup C_{2j+1}$ is a Möbius–Kantor configuration 8₃, cf. [20].

Taking into account ${\sigma }^{2i}\left(C_0\cup C_{2j+1}\right)=C_{2i(\mathrm{m}\mathrm{o}\mathrm{d}{\displaystyle \frac{q^2-1}{4}})}\cup C_{2i+2j+1(\mathrm{m}\mathrm{o}\mathrm{d}{\displaystyle \frac{q^2-1}{4}})}$, $i=0,1,2,\dots ,{\displaystyle \frac{q^2-9}{8}}$, we obtain the required partition. □

In the next sections, we present two examples. The first one shows that the Berardi–Masini mixed partition of AG(2,7) is cyclic, while the second one is focused on the role the cyclic structure of AG(2,19) plays in the construction developed in Theorem 1.

4. The Mixed Partition of AG(2,7)

Now, by explicitly constructing the cyclic structure of AG(2,7), we prove

Theorem 2.

Any affine plane AG(2,7) admits a partition into a set of Möbius–Kantor configurations 8₃ plus a set consisting only of one point.

Proof. 

Let us consider a primitive polynomial of degree two with minimal weight, i.e., the minimal number of nonzero coefficients, among all primitives of that degree over in ${\mathbb{F}}_7$, p(x) = x² + x + 3, cf. [21]. The element $3+4\sqrt{3}$ is a root of p(x) and then it is a primitive element of ${\mathbb{F}}_{7^2}$. Thus, the matrix $\left(\begin{array}{cc}3& 5\\ 4& 3\end{array}\right)$ induces a cycle $\sigma$ on the set of points (x,y) ≠ (0,0) of AG(2,7). If we choose the point (1,0) as base point, then $0={\sigma }^0\left(1,0\right)=\left(1,0\right);$ $1=\sigma \left(1,0\right)=\left(\begin{array}{cc}3& 5\\ 4& 3\end{array}\right)\left(\begin{array}{c}1\\ 0\end{array}\right)=\left(\begin{array}{c}3\\ 4\end{array}\right);$ $2={\sigma }^2\left(1,0\right)=\sigma \left(3,4\right)=\left(\begin{array}{cc}3& 5\\ 4& 3\end{array}\right)\left(\begin{array}{c}3\\ 4\end{array}\right)=\left(\begin{array}{c}1\\ 3\end{array}\right);$ by continuing in this way and by denoting the points represented by ${\sigma }^i\left(1,0\right)$ simply by $i$, the points of AG(2,7) can be identified with $\infty =\left(0,0\right)$ and the elements of the additive cyclic group ${\mathbb{Z}}_{48}$, the integers modulo 48, as shown in

Table 1. The cyclic structure of the points of AG(2,7).

$\mathit{\infty }=\left(0,0\right)$
0 = (1,0)	1 = (3,4)	2 = (1,3)	3 = (4,6)	4 = (0,6)	5 = (2,4)
6 = (5,6)	7 = (3,3)	8 = (3,0)	9 = (2,5)	10 = (3,2)	11 = (5,4)
12 = (0,4)	13 = (6,5)	14 = (1,4)	15 = (2,2)	16 = (2,0)	17 = (6,1)
18 = (2,6)	19 = (1,5)	20 = (0,5)	21 = (4,1)	22 = (3,5)	23 = (6,6)
24 = (6,0)	25 = (4,3)	26 = (6,4)	27 = (3,1)	28 = (0,1)	29 = (5,3)
30 = (2,1)	31 = (4,4)	32 = (4,0)	33 = (5,2)	34 = (4,5)	35 = (2,3)
36 = (0,3)	37 = (1,2)	38 = (6,3)	39 = (5,5)	40 = (5,0)	41 = (1,6)
42 = (5,1)	43 = (6,2)	44 = (0,2)	45 = (3,6)	46 = (4,2)	47 = (1,1)

. Actually, to write the table, it is not necessary to calculate all the values. It is sufficient to calculate only the first 8 values and see how the point (1,0) is transformed by ${\sigma }^8.$ In this case ${\sigma }^8\left(1,0\right)=\left(3,0\right)=3·\left(1,0\right).$ Thus, the next eight points are obtained from the previous ones by multiplying them by 3 modulo 7.

Select any line, for example, we choose the line ℓ₀:= x = 1, which contains the 7-set of points written in

Table 2. The starting line.

ℓ0

0

2

14

19

37

41

47

.

The remaining lines not containing the point $\infty$ of the plane are found by adding 1 to each point of the preceding line beginning with ℓ₀ and using addition modulo 48.

The 48 lines not containing the point $\infty$ of AG(2,7) consist of the points in the rows of

Table 3. The lines not containing the point $\infty$ of AG(2,7).

ℓ0	0	2	14	19	37	41	47
ℓ1	0	1	3	15	20	38	42
ℓ2	1	2	4	16	21	39	43
ℓ3	2	3	5	17	22	40	44
ℓ4	3	4	6	18	23	41	45
ℓ5	4	5	7	19	24	42	46
ℓ6	5	6	8	20	25	43	47
ℓ7	0	6	7	9	21	26	44
ℓ8	1	7	8	10	22	27	45
ℓ9	2	8	9	11	23	28	46
ℓ10	3	9	10	12	24	29	47
ℓ11	0	4	10	11	13	25	30
ℓ12	1	5	11	12	14	26	31
ℓ13	2	6	12	13	15	27	32
ℓ14	3	7	13	14	16	28	33
ℓ15	4	8	14	15	17	29	34
ℓ16	5	9	15	16	18	30	35
ℓ17	6	10	16	17	19	31	36
ℓ18	7	11	17	18	20	32	37
ℓ19	8	12	18	19	21	33	38
ℓ20	9	13	19	20	22	34	39
ℓ21	10	14	20	21	23	35	40
ℓ22	11	15	21	22	24	36	41
ℓ23	12	16	22	23	25	37	42
ℓ24	13	17	23	24	26	38	43
ℓ25	14	18	24	25	27	39	44
ℓ26	15	19	25	26	28	40	45
ℓ27	16	20	26	27	29	41	46
ℓ28	17	21	27	28	30	42	47
ℓ29	0	18	22	28	29	31	43
ℓ30	1	19	23	29	30	32	44
ℓ31	2	20	24	30	31	33	45
ℓ32	3	21	25	31	32	34	46
ℓ33	4	22	26	32	33	35	47
ℓ34	0	5	23	27	33	34	36
ℓ35	1	6	24	28	34	35	37
ℓ36	2	7	25	29	35	36	38
ℓ37	3	8	26	30	36	37	39
ℓ38	4	9	27	31	37	38	40
ℓ39	5	10	28	32	38	39	41
ℓ40	6	11	29	33	39	40	42
ℓ41	7	12	30	34	40	41	43
ℓ42	8	13	31	35	41	42	44
ℓ43	9	14	32	36	42	43	45
ℓ44	10	15	33	37	43	44	46
ℓ45	11	16	34	38	44	45	47
ℓ46	0	12	17	35	39	45	46
ℓ47	1	13	18	36	40	46	47

.

The lines containing the point $\infty$ of the plane are found by tacking the cosets of the unique subgroup of order 6 of ${\mathbb{Z}}_{48}$ written in

Table 4. The lines containing the point $\infty$ of AG(2,7).

m0	$\infty$	0	8	16	24	32	40
m1	$\infty$	1	9	17	25	33	41
m2	$\infty$	2	10	18	26	34	42
m3	$\infty$	3	11	19	27	35	43
m4	$\infty$	4	12	20	28	36	44
m5	$\infty$	5	13	21	29	37	45
m6	$\infty$	6	14	22	30	38	46
m7	$\infty$	7	15	23	31	39	47

.

The cosets of the unique subgroup of order 8 of ${\mathbb{Z}}_{48}$ define a partition in 6 conics of the set of points of AG(2,7)−$\left\{\infty \right\}$ written in

Table 5. The partition of AG(2,7)−$\left\{\infty \right\}$ into 6 conics.

C0	0	6	12	18	24	30	36	42
C1	1	7	13	19	25	31	37	43
C2	2	8	14	20	26	32	38	44
C3	3	9	15	21	27	33	39	45
C4	4	10	16	22	28	34	40	46
C5	5	11	17	23	29	35	41	47

.

The orbits of the points, different from $\infty$, under the unique subgroup of order 4 of ${\mathbb{Z}}_{48}$ define a partition into 12 quadrilaterals written in

Table 6. The partition of AG(2,7)−$\left\{\infty \right\}$ into 12 quadrilaterals.

C0	0	12	24	36
C1	1	13	25	37
C2	2	14	26	38
C3	3	15	27	39
C4	4	16	28	40
C5	5	17	29	41
C6	6	18	30	42
C7	7	19	31	43
C8	8	20	32	44
C9	9	21	33	45
C10	10	22	34	46
C11	11	23	35	47

.

Let us take into account C₀. The first quadrilateral C_i, i = 1,2,…,11, such that C₀ and C_i are simultaneously inscribed and circumscribed, as shown in Figure 1, is C₃.

Thus, the union C₀∪C₃ is the classical representation of the Möbius–Kantor configuration (8₃), cf. [20]. In Figure 2 below, from MathWorld, is the Möbius–Kantor configuration (8₃).

Finally, we prove the cyclic partition of AG(2,7). Since ${\sigma }^i\left(C_0\right)=C_i$, we have that ${\sigma }^0\left(C_0\cup C_3\right)=C_0\cup C_3$, ${\sigma }^2\left(C_0\cup C_3\right)=C_2\cup C_5$, ${\sigma }^4\left(C_0\cup C_3\right)={\sigma }^2\left(C_2\cup C_5\right)=C_4\cup C_7$, ${\sigma }^6\left(C_0\cup C_3\right)={\sigma }^2\left(C_4\cup C_7\right)=C_6\cup C_9$, ${{\sigma }^8\left(C_0\cup C_3\right)=\sigma }^2\left(C_6\cup C_9\right)=C_8\cup C_{11}$ and ${\sigma }^{10}\left(C_0\cup C_3\right){=\sigma }^2\left(C_8\cup C_{11}\right)=C_1\cup C_{10}$. Therefore, we obtain the cyclic partition of AG(2,7), shown in Figure 3, cf. [9]. □

5. The Mixed Partition of AG(2,19)

Now, by explicitly constructing the cyclic structure of AG(2,19), we prove

Theorem 3.

Any affine plane AG(2,19) admits a partition into a set of Möbius–Kantor configurations 8₃ plus a set consisting only of one point.

Proof. 

Let us consider a primitive polynomial of degree two with minimal weight, i.e., the minimal number of nonzero coefficients, among all primitives of that degree over in ${\mathbb{F}}_{19}$, p(x) = x² + x + 2, cf. [21]. It is a simple exercise to verify that the element $9+10\sqrt{12}$ is a root of p(x) and then a primitive element of ${\mathbb{F}}_{{19}^2}$. Thus, the matrix $\left(\begin{array}{cc}9& 6\\ 10& 9\end{array}\right)$ induces a cycle $\sigma$ on the set of points (x,y) ≠ (0,0) of AG(2,19). If we choose the point (1,0) as a base point, then $0={\sigma }^0\left(1,0\right)=\left(1,0\right);$ $1=\sigma \left(1,0\right)=\left(\begin{array}{cc}9& 6\\ 10& 9\end{array}\right)\left(\begin{array}{c}1\\ 0\end{array}\right)=\left(\begin{array}{c}9\\ 10\end{array}\right);$ $2={\sigma }^2\left(1,0\right)=\sigma \left(9,10\right)=\left(\begin{array}{cc}9& 6\\ 10& 9\end{array}\right)\left(\begin{array}{c}9\\ 10\end{array}\right)=\left(\begin{array}{c}8\\ 9\end{array}\right);$ by continuing in this way and by denoting the points represented by ${\sigma }^i\left(1,0\right)$ simply by $i$, the points of AG(2,19) can be identified with $\infty =\left(0,0\right)$ and the elements of the additive cyclic group ${\mathbb{Z}}_{360}$, the integers modulo 360, as shown in

Table A1. The cyclic structure of the points of AG(2,19).

$\mathit{\infty }=\left(0,0\right)$
0 = (1,0)	1 = (9,10)	2 = (8,9)	3 = (12,9)	4 = (10,11)	5 = (4,9)
6 = (14,7)	7 = (16,13)	8 = (13,11)	9 = (12,1)	10 = (0,15)	11 = (14,2)
12 = (5,6)	13 = (5,9)	14 = (4,17)	15 = (5,3)	16 = (6,1)	17 = (3,12)
18 = (4,5)	19 = (9,9)	20 = (2,0)	21 = (18,1)	22 = (16,18)	23 = (5,18)
24 = (1,3)	25 = (8,18)	26 = (9,14)	27 = (13,7)	28 = (7,3)	29 = (5,2)
30 = (0,11)	31 = (9,4)	32 = (10,12)	33 = (10,18)	34 = (8,15)	35 = (10,6)
36 = (12,2)	37 = (6,5)	38 = (8,10)	39 = (18,18)	40 = (4,0)	41 = (17,2)
42 = (13,17)	43 = (10,17)	44 = (2,6)	45 = (16,17)	46 = (18,9)	47 = (7,14)
48 = (14,6)	49 = (10,4)	50 = (0,3)	51 = (18,8)	52 = (1,5)	53 = (1,17)
54 = (16,11)	55 = (1,12)	56 = (5,4)	57 = (12,10)	58 = (16,1)	59 = (17,17)
60 = (8,0)	61 = (15,4)	62 = (7,15)	63 = (1,15)	64 = (4,12)	65 = (13,15)
66 = (17,18)	67 = (14,9)	68 = (9,12)	69 = (1,8)	70 = (0,6)	71 = (17,16)
72 = (2,10)	73 = (2,15)	74 = (13,3)	75 = (2,5)	76 = (10,8)	77 = (5,1)
78 = (13,2)	79 = (15,15)	80 = (16,0)	81 = (11,8)	82 = (14,11)	83 = (2,11)
84 = (8,5)	85 = (7,11)	86 = (15,17)	87 = (9,18)	88 = (18,5)	89 = (2,16)
90 = (0,12)	91 = (15,13)	92 = (4,1)	93 = (4,11)	94 = (7,6)	95 = (4,10)
96 = (1,16)	97 = (10,2)	98 = (7,4)	99 = (11,11)	100 = (13,0)	101 = (3,16)
102 = (9,3)	103 = (4,3)	104 = (16,10)	105 = (14,3)	106 = (11,15)	107 = (18,17)
108 = (17,10)	109 = (4,13)	110 = (0,5)	111 = (11,7)	112 = (8,2)	113 = (8,3)
114 = (14,12)	115 = (8,1)	116 = (2,13)	117 = (1,4)	118 = (14,8)	119 = (3,3)
120 = (7,0)	121 = (6,13)	122 = (18,6)	123 = (8,6)	124 = (13,1)	125 = (9,6)
126 = (3,11)	127 = (17,15)	128 = (15,1)	129 = (8,7)	130 = (0,10)	131 = (3,14)
132 = (16,4)	133 = (16,6)	134 = (9,5)	135 = (16,2)	136 = (4,7)	137 = (2,8)
138 = (9,16)	139 = (6,6)	140 = (14,0)	141 = (12,7)	142 = (17,12)	143 = (16,12)
144 = (7,2)	145 = (18,12)	146 = (6,3)	147 = (15,11)	148 = (11,2)	149 = (16,14)
150 = (0,1)	151 = (6,9)	152 = (13,8)	153 = (13,12)	154 = (18,10)	155 = (13,4)
156 = (8,14)	157 = (4,16)	158 = (18,13)	159 = (12,12)	160 = (9,0)	161 = (5,14)
162 = (15,5)	163 = (13,5)	164 = (14,4)	165 = (17,5)	166 = (12,6)	167 = (11,3)
168 = (3,4)	169 = (13,9)	170 = (0,2)	171 = (12,18)	172 = (7,16)	173 = (7,5)
174 = (17,1)	175 = (7,8)	176 = (16,9)	177 = (8,13)	178 = (17,7)	179 = (5,5)
180 = (18,0)	181 = (10,9)	182 = (11,10)	183 = (7,10)	184 = (9,8)	185 = (15,10)
186 = (5,12)	187 = (3,6)	188 = (6,8)	189 = (7,18)	190 = (0,4)	191 = (5,17)
192 = (14,13)	193 = (14,10)	194 = (15,2)	195 = (14,16)	196 = (13,18)	197 = (16,7)
198 = (15,14)	199 = (10,10)	200 = (17,0)	201 = (1,18)	202 = (3,1)	203 = (14,1)
204 = (18,16)	205 = (11,1)	206 = (10,5)	207 = (6,12)	208 = (12,16)	209 = (14,17)
210 = (0,8)	211 = (10,15)	212 = (9,7)	213 = (9,1)	214 = (11,4)	215 = (9,13)
216 = (7,17)	217 = (13,14)	218 = (11,9)	219 = (1,1)	220 = (15,0)	221 = (2,17)
222 = (6,2)	223 = (9,2)	224 = (17,13)	225 = (3,2)	226 = (1,10)	227 = (12,5)
228 = (5,13)	229 = (9,15)	230 = (0,16)	231 = (1,11)	232 = (18,14)	233 = (18,2)
234 = (3,8)	235 = (18,7)	236 = (14,15)	237 = (7,9)	238 = (3,18)	239 = (2,2)
240 = (11,0)	241 = (4,15)	242 = (12,4)	243 = (18,4)	244 = (15,7)	245 = (6,4)
246 = (2,1)	247 = (5,10)	248 = (10,7)	249 = (18,11)	250 = (0,13)	251 = (2,3)
252 = (17,9)	253 = (17,4)	254 = (6,16)	255 = (17,14)	256 = (9,11)	257 = (14,18)
258 = (6,17)	259 = (4,4)	260 = (3,0)	261 = (8,11)	262 = (5,8)	263 = (17,8)
264 = (11,14)	265 = (12,8)	266 = (4,2)	267 = (10,1)	268 = (1,14)	269 = (17,3)
270 = (0,7)	271 = (4,6)	272 = (15,18)	273 = (15,8)	274 = (12,13)	275 = (15,9)
276 = (18,3)	277 = (9,17)	278 = (12,15)	279 = (8,8)	280 = (6,0)	281 = (16,3)
282 = (10,16)	283 = (15,16)	284 = (3,9)	285 = (5,16)	286 = (8,4)	287 = (1,2)
288 = (2,9)	289 = (15,6)	290 = (0,14)	291 = (8,12)	292 = (11,17)	293 = (11,16)
294 = (5,7)	295 = (11,18)	296 = (17,6)	297 = (18,15)	298 = (5,11)	299 = (16,16)
300 = (12,0)	301 = (13,6)	302 = (1,13)	303 = (11,13)	304 = (6,18)	305 = (10,13)
306 = (16,8)	307 = (2,4)	308 = (4,18)	309 = (11,12)	310 = (0,9)	311 = (16,5)
312 = (3,15)	313 = (3,13)	314 = (10,14)	315 = (3,17)	316 = (15,12)	317 = (17,11)
318 = (10,3)	319 = (13,13)	320 = (5,0)	321 = (7,12)	322 = (2,7)	323 = (3,7)
324 = (12,17)	325 = (1,7)	326 = (13,16)	327 = (4,8)	328 = (8,17)	329 = (3,5)
330 = (0,18)	331 = (13,10)	332 = (6,11)	333 = (6,7)	334 = (1,9)	335 = (6,15)
336 = (11,5)	337 = (15,3)	338 = (1,6)	339 = (7,7)	340 = (10,0)	341 = (14,5)
342 = (4,14)	343 = (6,14)	344 = (5,15)	345 = (2,14)	346 = (7,13)	347 = (8,16)
348 = (16,15)	349 = (6,10)	350 = (0,17)	351 = (7,1)	352 = (12,3)	353 = (12,14)
354 = (2,18)	355 = (12,11)	356 = (3,10)	357 = (11,6)	358 = (2,12)	359 = (14,14)

. Actually, to write the table, it is not necessary to calculate all the values. It is sufficient to calculate only the first 20 values and see how the point (1,0) is transformed by ${\sigma }^{20}.$ In this case, ${\sigma }^{20}\left(1,0\right)=\left(2,0\right)=2·\left(1,0\right).$ Thus, the next twenty points are obtained from the previous ones by multiplying them by 2 modulo 19. To enhance the readability of the paper, the cyclic structure of the points of AG(2,19) is placed as Table A1 in Appendix A.

Select any line, for example, we choose the line ℓ₀ := x = 1, which contains the 19-set of points written in

Table 7. The starting line.

ℓ0

0

24

52

53

55

63

69

96

117

201

219

226

231

268

287

302

325

334

338

.

The remaining lines not containing the point $\infty$ of the plane are found by adding 1 to each point of the preceding line beginning with ℓ₀ and using addition modulo 360.

The 360 lines not containing the point $\infty$ of AG(2,19) consist of the points in the rows of

Table A2. The 360 lines not containing the point $\infty$ of AG(2,19).

ℓ0	0	24	52	53	55	63	69	96	117	201	219	226	231	268	287	302	325	334	338
ℓ1	1	25	53	54	56	64	70	97	118	202	220	227	232	269	288	303	326	335	339
ℓ2	2	26	54	55	57	65	71	98	119	203	221	228	233	270	289	304	327	336	340
ℓ3	3	27	55	56	58	66	72	99	120	204	222	229	234	271	290	305	328	337	341
ℓ4	4	28	56	57	59	67	73	100	121	205	223	230	235	272	291	306	329	338	342
ℓ5	5	29	57	58	60	68	74	101	122	206	224	231	236	273	292	307	330	339	343
ℓ6	6	30	58	59	61	69	75	102	123	207	225	232	237	274	293	308	331	340	344
ℓ7	7	31	59	60	62	70	76	103	124	208	226	233	238	275	294	309	332	341	345
ℓ8	8	32	60	61	63	71	77	104	125	209	227	234	239	276	295	310	333	342	346
ℓ9	9	33	61	62	64	72	78	105	126	210	228	235	240	277	296	311	334	343	347
ℓ10	10	34	62	63	65	73	79	106	127	211	229	236	241	278	297	312	335	344	348
ℓ11	11	35	63	64	66	74	80	107	128	212	230	237	242	279	298	313	336	345	349
ℓ12	12	36	64	65	67	75	81	108	129	213	231	238	243	280	299	314	337	346	350
ℓ13	13	37	65	66	68	76	82	109	130	214	232	239	244	281	300	315	338	347	351
ℓ14	14	38	66	67	69	77	83	110	131	215	233	240	245	282	301	316	339	348	352
ℓ15	15	39	67	68	70	78	84	111	132	216	234	241	246	283	302	317	340	349	353
ℓ16	16	40	68	69	71	79	85	112	133	217	235	242	247	284	303	318	341	350	354
ℓ17	17	41	69	70	72	80	86	113	134	218	236	243	248	285	304	319	342	351	355
ℓ18	18	42	70	71	73	81	87	114	135	219	237	244	249	286	305	320	343	352	356
ℓ19	19	43	71	72	74	82	88	115	136	220	238	245	250	287	306	321	344	353	357
ℓ20	20	44	72	73	75	83	89	116	137	221	239	246	251	288	307	322	345	354	358
ℓ21	21	45	73	74	76	84	90	117	138	222	240	247	252	289	308	323	346	355	359
ℓ22	0	22	46	74	75	77	85	91	118	139	223	241	248	253	290	309	324	347	356
ℓ23	1	23	47	75	76	78	86	92	119	140	224	242	249	254	291	310	325	348	357
ℓ24	2	24	48	76	77	79	87	93	120	141	225	243	250	255	292	311	326	349	358
ℓ25	3	25	49	77	78	80	88	94	121	142	226	244	251	256	293	312	327	350	359
ℓ26	0	4	26	50	78	79	81	89	95	122	143	227	245	252	257	294	313	328	351
ℓ27	1	5	27	51	79	80	82	90	96	123	144	228	246	253	258	295	314	329	352
ℓ28	2	6	28	52	80	81	83	91	97	124	145	229	247	254	259	296	315	330	353
ℓ29	3	7	29	53	81	82	84	92	98	125	146	230	248	255	260	297	316	331	354
ℓ30	4	8	30	54	82	83	85	93	99	126	147	231	249	256	261	298	317	332	355
ℓ31	5	9	31	55	83	84	86	94	100	127	148	232	250	257	262	299	318	333	356
ℓ32	6	10	32	56	84	85	87	95	101	128	149	233	251	258	263	300	319	334	357
ℓ33	7	11	33	57	85	86	88	96	102	129	150	234	252	259	264	301	320	335	358
ℓ34	8	12	34	58	86	87	89	97	103	130	151	235	253	260	265	302	321	336	359
ℓ35	0	9	13	35	59	87	88	90	98	104	131	152	236	254	261	266	303	322	337
ℓ36	1	10	14	36	60	88	89	91	99	105	132	153	237	255	262	267	304	323	338
ℓ37	2	11	15	37	61	89	90	92	100	106	133	154	238	256	263	268	305	324	339
ℓ38	3	12	16	38	62	90	91	93	101	107	134	155	239	257	264	269	306	325	340
ℓ39	4	13	17	39	63	91	92	94	102	108	135	156	240	258	265	270	307	326	341
ℓ40	5	14	18	40	64	92	93	95	103	109	136	157	241	259	266	271	308	327	342
ℓ41	6	15	19	41	65	93	94	96	104	110	137	158	242	260	267	272	309	328	343
ℓ42	7	16	20	42	66	94	95	97	105	111	138	159	243	261	268	273	310	329	344
ℓ43	8	17	21	43	67	95	96	98	106	112	139	160	244	262	269	274	311	330	345
ℓ44	9	18	22	44	68	96	97	99	107	113	140	161	245	263	270	275	312	331	346
ℓ45	10	19	23	45	69	97	98	100	108	114	141	162	246	264	271	276	313	332	347
ℓ46	11	20	24	46	70	98	99	101	109	115	142	163	247	265	272	277	314	333	348
ℓ47	12	21	25	47	71	99	100	102	110	116	143	164	248	266	273	278	315	334	349
ℓ48	13	22	26	48	72	100	101	103	111	117	144	165	249	267	274	279	316	335	350
ℓ49	14	23	27	49	73	101	102	104	112	118	145	166	250	268	275	280	317	336	351
ℓ50	15	24	28	50	74	102	103	105	113	119	146	167	251	269	276	281	318	337	352
ℓ51	16	25	29	51	75	103	104	106	114	120	147	168	252	270	277	282	319	338	353
ℓ52	17	26	30	52	76	104	105	107	115	121	148	169	253	271	278	283	320	339	354
ℓ53	18	27	31	53	77	105	106	108	116	122	149	170	254	272	279	284	321	340	355
ℓ54	19	28	32	54	78	106	107	109	117	123	150	171	255	273	280	285	322	341	356
ℓ55	20	29	33	55	79	107	108	110	118	124	151	172	256	274	281	286	323	342	357
ℓ56	21	30	34	56	80	108	109	111	119	125	152	173	257	275	282	287	324	343	358
ℓ57	22	31	35	57	81	109	110	112	120	126	153	174	258	276	283	288	325	344	359
ℓ58	0	23	32	36	58	82	110	111	113	121	127	154	175	259	277	284	289	326	345
ℓ59	1	24	33	37	59	83	111	112	114	122	128	155	176	260	278	285	290	327	346
ℓ60	2	25	34	38	60	84	112	113	115	123	129	156	177	261	279	286	291	328	347
ℓ61	3	26	35	39	61	85	113	114	116	124	130	157	178	262	280	287	292	329	348
ℓ62	4	27	36	40	62	86	114	115	117	125	131	158	179	263	281	288	293	330	349
ℓ63	5	28	37	41	63	87	115	116	118	126	132	159	180	264	282	289	294	331	350
ℓ64	6	29	38	42	64	88	116	117	119	127	133	160	181	265	283	290	295	332	351
ℓ65	7	30	39	43	65	89	117	118	120	128	134	161	182	266	284	291	296	333	352
ℓ66	8	31	40	44	66	90	118	119	121	129	135	162	183	267	285	292	297	334	353
ℓ67	9	32	41	45	67	91	119	120	122	130	136	163	184	268	286	293	298	335	354
ℓ68	10	33	42	46	68	92	120	121	123	131	137	164	185	269	287	294	299	336	355
ℓ69	11	34	43	47	69	93	121	122	124	132	138	165	186	270	288	295	300	337	356
ℓ70	12	35	44	48	70	94	122	123	125	133	139	166	187	271	289	296	301	338	357
ℓ71	13	36	45	49	71	95	123	124	126	134	140	167	188	272	290	297	302	339	358
ℓ72	14	37	46	50	72	96	124	125	127	135	141	168	189	273	291	298	303	340	359
ℓ73	0	15	38	47	51	73	97	125	126	128	136	142	169	190	274	292	299	304	341
ℓ74	1	16	39	48	52	74	98	126	127	129	137	143	170	191	275	293	300	305	342
ℓ75	2	17	40	49	53	75	99	127	128	130	138	144	171	192	276	294	301	306	343
ℓ76	3	18	41	50	54	76	100	128	129	131	139	145	172	193	277	295	302	307	344
ℓ77	4	19	42	51	55	77	101	129	130	132	140	146	173	194	278	296	303	308	345
ℓ78	5	20	43	52	56	78	102	130	131	133	141	147	174	195	279	297	304	309	346
ℓ79	6	21	44	53	57	79	103	131	132	134	142	148	175	196	280	298	305	310	347
ℓ80	7	22	45	54	58	80	104	132	133	135	143	149	176	197	281	299	306	311	348
ℓ81	8	23	46	55	59	81	105	133	134	136	144	150	177	198	282	300	307	312	349
ℓ82	9	24	47	56	60	82	106	134	135	137	145	151	178	199	283	301	308	313	350
ℓ83	10	25	48	57	61	83	107	135	136	138	146	152	179	200	284	302	309	314	351
ℓ84	11	26	49	58	62	84	108	136	137	139	147	153	180	201	285	303	310	315	352
ℓ85	12	27	50	59	63	85	109	137	138	140	148	154	181	202	286	304	311	316	353
ℓ86	13	28	51	60	64	86	110	138	139	141	149	155	182	203	287	305	312	317	354
ℓ87	14	29	52	61	65	87	111	139	140	142	150	156	183	204	288	306	313	318	355
ℓ88	15	30	53	62	66	88	112	140	141	143	151	157	184	205	289	307	314	319	356
ℓ89	16	31	54	63	67	89	113	141	142	144	152	158	185	206	290	308	315	320	357
ℓ90	17	32	55	64	68	90	114	142	143	145	153	159	186	207	291	309	316	321	358
ℓ91	18	33	56	65	69	91	115	143	144	146	154	160	187	208	292	310	317	322	359
ℓ92	0	19	34	57	66	70	92	116	144	145	147	155	161	188	209	293	311	318	323
ℓ93	1	20	35	58	67	71	93	117	145	146	148	156	162	189	210	294	312	319	324
ℓ94	2	21	36	59	68	72	94	118	146	147	149	157	163	190	211	295	313	320	325
ℓ95	3	22	37	60	69	73	95	119	147	148	150	158	164	191	212	296	314	321	326
ℓ96	4	23	38	61	70	74	96	120	148	149	151	159	165	192	213	297	315	322	327
ℓ97	5	24	39	62	71	75	97	121	149	150	152	160	166	193	214	298	316	323	328
ℓ98	6	25	40	63	72	76	98	122	150	151	153	161	167	194	215	299	317	324	329
ℓ99	7	26	41	64	73	77	99	123	151	152	154	162	168	195	216	300	318	325	330
ℓ100	8	27	42	65	74	78	100	124	152	153	155	163	169	196	217	301	319	326	331
ℓ101	9	28	43	66	75	79	101	125	153	154	156	164	170	197	218	302	320	327	332
ℓ102	10	29	44	67	76	80	102	126	154	155	157	165	171	198	219	303	321	328	333
ℓ103	11	30	45	68	77	81	103	127	155	156	158	166	172	199	220	304	322	329	334
ℓ104	12	31	46	69	78	82	104	128	156	157	159	167	173	200	221	305	323	330	335
ℓ105	13	32	47	70	79	83	105	129	157	158	160	168	174	201	222	306	324	331	336
ℓ106	14	33	48	71	80	84	106	130	158	159	161	169	175	202	223	307	325	332	337
ℓ107	15	34	49	72	81	85	107	131	159	160	162	170	176	203	224	308	326	333	338
ℓ108	16	35	50	73	82	86	108	132	160	161	163	171	177	204	225	309	327	334	339
ℓ109	17	36	51	74	83	87	109	133	161	162	164	172	178	205	226	310	328	335	340
ℓ110	18	37	52	75	84	88	110	134	162	163	165	173	179	206	227	311	329	336	341
ℓ111	19	38	53	76	85	89	111	135	163	164	166	174	180	207	228	312	330	337	342
ℓ112	20	39	54	77	86	90	112	136	164	165	167	175	181	208	229	313	331	338	343
ℓ113	21	40	55	78	87	91	113	137	165	166	168	176	182	209	230	314	332	339	344
ℓ114	22	41	56	79	88	92	114	138	166	167	169	177	183	210	231	315	333	340	345
ℓ115	23	42	57	80	89	93	115	139	167	168	170	178	184	211	232	316	334	341	346
ℓ116	24	43	58	81	90	94	116	140	168	169	171	179	185	212	233	317	335	342	347
ℓ117	25	44	59	82	91	95	117	141	169	170	172	180	186	213	234	318	336	343	348
ℓ118	26	45	60	83	92	96	118	142	170	171	173	181	187	214	235	319	337	344	349
ℓ119	27	46	61	84	93	97	119	143	171	172	174	182	188	215	236	320	338	345	350
ℓ120	28	47	62	85	94	98	120	144	172	173	175	183	189	216	237	321	339	346	351
ℓ121	29	48	63	86	95	99	121	145	173	174	176	184	190	217	238	322	340	347	352
ℓ122	30	49	64	87	96	100	122	146	174	175	177	185	191	218	239	323	341	348	353
ℓ123	31	50	65	88	97	101	123	147	175	176	178	186	192	219	240	324	342	349	354
ℓ124	32	51	66	89	98	102	124	148	176	177	179	187	193	220	241	325	343	350	355
ℓ125	33	52	67	90	99	103	125	149	177	178	180	188	194	221	242	326	344	351	356
ℓ126	34	53	68	91	100	104	126	150	178	179	181	189	195	222	243	327	345	352	357
ℓ127	35	54	69	92	101	105	127	151	179	180	182	190	196	223	244	328	346	353	358
ℓ128	36	55	70	93	102	106	128	152	180	181	183	191	197	224	245	329	347	354	359
ℓ129	0	37	56	71	94	103	107	129	153	181	182	184	192	198	225	246	330	348	355
ℓ130	1	38	57	72	95	104	108	130	154	182	183	185	193	199	226	247	331	349	356
ℓ131	2	39	58	73	96	105	109	131	155	183	184	186	194	200	227	248	332	350	357
ℓ132	3	40	59	74	97	106	110	132	156	184	185	187	195	201	228	249	333	351	358
ℓ133	4	41	60	75	98	107	111	133	157	185	186	188	196	202	229	250	334	352	359
ℓ134	0	5	42	61	76	99	108	112	134	158	186	187	189	197	203	230	251	335	353
ℓ135	1	6	43	62	77	100	109	113	135	159	187	188	190	198	204	231	252	336	354
ℓ136	2	7	44	63	78	101	110	114	136	160	188	189	191	199	205	232	253	337	355
ℓ137	3	8	45	64	79	102	111	115	137	161	189	190	192	200	206	233	254	338	356
ℓ138	4	9	46	65	80	103	112	116	138	162	190	191	193	201	207	234	255	339	357
ℓ139	5	10	47	66	81	104	113	117	139	163	191	192	194	202	208	235	256	340	358
ℓ140	6	11	48	67	82	105	114	118	140	164	192	193	195	203	209	236	257	341	359
ℓ141	0	7	12	49	68	83	106	115	119	141	165	193	194	196	204	210	237	258	342
ℓ142	1	8	13	50	69	84	107	116	120	142	166	194	195	197	205	211	238	259	343
ℓ143	2	9	14	51	70	85	108	117	121	143	167	195	196	198	206	212	239	260	344
ℓ144	3	10	15	52	71	86	109	118	122	144	168	196	197	199	207	213	240	261	345
ℓ145	4	11	16	53	72	87	110	119	123	145	169	197	198	200	208	214	241	262	346
ℓ146	5	12	17	54	73	88	111	120	124	146	170	198	199	201	209	215	242	263	347
ℓ147	6	13	18	55	74	89	112	121	125	147	171	199	200	202	210	216	243	264	348
ℓ148	7	14	19	56	75	90	113	122	126	148	172	200	201	203	211	217	244	265	349
ℓ149	8	15	20	57	76	91	114	123	127	149	173	201	202	204	212	218	245	266	350
ℓ150	9	16	21	58	77	92	115	124	128	150	174	202	203	205	213	219	246	267	351
ℓ151	10	17	22	59	78	93	116	125	129	151	175	203	204	206	214	220	247	268	352
ℓ152	11	18	23	60	79	94	117	126	130	152	176	204	205	207	215	221	248	269	353
ℓ153	12	19	24	61	80	95	118	127	131	153	177	205	206	208	216	222	249	270	354
ℓ154	13	20	25	62	81	96	119	128	132	154	178	206	207	209	217	223	250	271	355
ℓ155	14	21	26	63	82	97	120	129	133	155	179	207	208	210	218	224	251	272	356
ℓ156	15	22	27	64	83	98	121	130	134	156	180	208	209	211	219	225	252	273	357
ℓ157	16	23	28	65	84	99	122	131	135	157	181	209	210	212	220	226	253	274	358
ℓ158	17	24	29	66	85	100	123	132	136	158	182	210	211	213	221	227	254	275	359
ℓ159	0	18	25	30	67	86	101	124	133	137	159	183	211	212	214	222	228	255	276
ℓ160	1	19	26	31	68	87	102	125	134	138	160	184	212	213	215	223	229	256	277
ℓ161	2	20	27	32	69	88	103	126	135	139	161	185	213	214	216	224	230	257	278
ℓ162	3	21	28	33	70	89	104	127	136	140	162	186	214	215	217	225	231	258	279
ℓ163	4	22	29	34	71	90	105	128	137	141	163	187	215	216	218	226	232	259	280
ℓ164	5	23	30	35	72	91	106	129	138	142	164	188	216	217	219	227	233	260	281
ℓ165	6	24	31	36	73	92	107	130	139	143	165	189	217	218	220	228	234	261	282
ℓ166	7	25	32	37	74	93	108	131	140	144	166	190	218	219	221	229	235	262	283
ℓ167	8	26	33	38	75	94	109	132	141	145	167	191	219	220	222	230	236	263	284
ℓ168	9	27	34	39	76	95	110	133	142	146	168	192	220	221	223	231	237	264	285
ℓ169	10	28	35	40	77	96	111	134	143	147	169	193	221	222	224	232	238	265	286
ℓ170	11	29	36	41	78	97	112	135	144	148	170	194	222	223	225	233	239	266	287
ℓ171	12	30	37	42	79	98	113	136	145	149	171	195	223	224	226	234	240	267	288
ℓ172	13	31	38	43	80	99	114	137	146	150	172	196	224	225	227	235	241	268	289
ℓ173	14	32	39	44	81	100	115	138	147	151	173	197	225	226	228	236	242	269	290
ℓ174	15	33	40	45	82	101	116	139	148	152	174	198	226	227	229	237	243	270	291
ℓ175	16	34	41	46	83	102	117	140	149	153	175	199	227	228	230	238	244	271	292
ℓ176	17	35	42	47	84	103	118	141	150	154	176	200	228	229	231	239	245	272	293
ℓ177	18	36	43	48	85	104	119	142	151	155	177	201	229	230	232	240	246	273	294
ℓ178	19	37	44	49	86	105	120	143	152	156	178	202	230	231	233	241	247	274	295
ℓ179	20	38	45	50	87	106	121	144	153	157	179	203	231	232	234	242	248	275	296
ℓ180	21	39	46	51	88	107	122	145	154	158	180	204	232	233	235	243	249	276	297
ℓ181	22	40	47	52	89	108	123	146	155	159	181	205	233	234	236	244	250	277	298
ℓ182	23	41	48	53	90	109	124	147	156	160	182	206	234	235	237	245	251	278	299
ℓ183	24	42	49	54	91	110	125	148	157	161	183	207	235	236	238	246	252	279	300
ℓ184	25	43	50	55	92	111	126	149	158	162	184	208	236	237	239	247	253	280	301
ℓ185	26	44	51	56	93	112	127	150	159	163	185	209	237	238	240	248	254	281	302
ℓ186	27	45	52	57	94	113	128	151	160	164	186	210	238	239	241	249	255	282	303
ℓ187	28	46	53	58	95	114	129	152	161	165	187	211	239	240	242	250	256	283	304
ℓ188	29	47	54	59	96	115	130	153	162	166	188	212	240	241	243	251	257	284	305
ℓ189	30	48	55	60	97	116	131	154	163	167	189	213	241	242	244	252	258	285	306
ℓ190	31	49	56	61	98	117	132	155	164	168	190	214	242	243	245	253	259	286	307
ℓ191	32	50	57	62	99	118	133	156	165	169	191	215	243	244	246	254	260	287	308
ℓ192	33	51	58	63	100	119	134	157	166	170	192	216	244	245	247	255	261	288	309
ℓ193	34	52	59	64	101	120	135	158	167	171	193	217	245	246	248	256	262	289	310
ℓ194	35	53	60	65	102	121	136	159	168	172	194	218	246	247	249	257	263	290	311
ℓ195	36	54	61	66	103	122	137	160	169	173	195	219	247	248	250	258	264	291	312
ℓ196	37	55	62	67	104	123	138	161	170	174	196	220	248	249	251	259	265	292	313
ℓ197	38	56	63	68	105	124	139	162	171	175	197	221	249	250	252	260	266	293	314
ℓ198	39	57	64	69	106	125	140	163	172	176	198	222	250	251	253	261	267	294	315
ℓ199	40	58	65	70	107	126	141	164	173	177	199	223	251	252	254	262	268	295	316
ℓ200	41	59	66	71	108	127	142	165	174	178	200	224	252	253	255	263	269	296	317
ℓ201	42	60	67	72	109	128	143	166	175	179	201	225	253	254	256	264	270	297	318
ℓ202	43	61	68	73	110	129	144	167	176	180	202	226	254	255	257	265	271	298	319
ℓ203	44	62	69	74	111	130	145	168	177	181	203	227	255	256	258	266	272	299	320
ℓ204	45	63	70	75	112	131	146	169	178	182	204	228	256	257	259	267	273	300	321
ℓ205	46	64	71	76	113	132	147	170	179	183	205	229	257	258	260	268	274	301	322
ℓ206	47	65	72	77	114	133	148	171	180	184	206	230	258	259	261	269	275	302	323
ℓ207	48	66	73	78	115	134	149	172	181	185	207	231	259	260	262	270	276	303	324
ℓ208	49	67	74	79	116	135	150	173	182	186	208	232	260	261	263	271	277	304	325
ℓ209	50	68	75	80	117	136	151	174	183	187	209	233	261	262	264	272	278	305	326
ℓ210	51	69	76	81	118	137	152	175	184	188	210	234	262	263	265	273	279	306	327
ℓ211	52	70	77	82	119	138	153	176	185	189	211	235	263	264	266	274	280	307	328
ℓ212	53	71	78	83	120	139	154	177	186	190	212	236	264	265	267	275	281	308	329
ℓ213	54	72	79	84	121	140	155	178	187	191	213	237	265	266	268	276	282	309	330
ℓ214	55	73	80	85	122	141	156	179	188	192	214	238	266	267	269	277	283	310	331
ℓ215	56	74	81	86	123	142	157	180	189	193	215	239	267	268	270	278	284	311	332
ℓ216	57	75	82	87	124	143	158	181	190	194	216	240	268	269	271	279	285	312	333
ℓ217	58	76	83	88	125	144	159	182	191	195	217	241	269	270	272	280	286	313	334
ℓ218	59	77	84	89	126	145	160	183	192	196	218	242	270	271	273	281	287	314	335
ℓ219	60	78	85	90	127	146	161	184	193	197	219	243	271	272	274	282	288	315	336
ℓ220	61	79	86	91	128	147	162	185	194	198	220	244	272	273	275	283	289	316	337
ℓ221	62	80	87	92	129	148	163	186	195	199	221	245	273	274	276	284	290	317	338
ℓ222	63	81	88	93	130	149	164	187	196	200	222	246	274	275	277	285	291	318	339
ℓ223	64	82	89	94	131	150	165	188	197	201	223	247	275	276	278	286	292	319	340
ℓ224	65	83	90	95	132	151	166	189	198	202	224	248	276	277	279	287	293	320	341
ℓ225	66	84	91	96	133	152	167	190	199	203	225	249	277	278	280	288	294	321	342
ℓ226	67	85	92	97	134	153	168	191	200	204	226	250	278	279	281	289	295	322	343
ℓ227	68	86	93	98	135	154	169	192	201	205	227	251	279	280	282	290	296	323	344
ℓ228	69	87	94	99	136	155	170	193	202	206	228	252	280	281	283	291	297	324	345
ℓ229	70	88	95	100	137	156	171	194	203	207	229	253	281	282	284	292	298	325	346
ℓ230	71	89	96	101	138	157	172	195	204	208	230	254	282	283	285	293	299	326	347
ℓ231	72	90	97	102	139	158	173	196	205	209	231	255	283	284	286	294	300	327	348
ℓ232	73	91	98	103	140	159	174	197	206	210	232	256	284	285	287	295	301	328	349
ℓ233	74	92	99	104	141	160	175	198	207	211	233	257	285	286	288	296	302	329	350
ℓ234	75	93	100	105	142	161	176	199	208	212	234	258	286	287	289	297	303	330	351
ℓ235	76	94	101	106	143	162	177	200	209	213	235	259	287	288	290	298	304	331	352
ℓ236	77	95	102	107	144	163	178	201	210	214	236	260	288	289	291	299	305	332	353
ℓ237	78	96	103	108	145	164	179	202	211	215	237	261	289	290	292	300	306	333	354
ℓ238	79	97	104	109	146	165	180	203	212	216	238	262	290	291	293	301	307	334	355
ℓ239	80	98	105	110	147	166	181	204	213	217	239	263	291	292	294	302	308	335	356
ℓ240	81	99	106	111	148	167	182	205	214	218	240	264	292	293	295	303	309	336	357
ℓ241	82	100	107	112	149	168	183	206	215	219	241	265	293	294	296	304	310	337	358
ℓ242	83	101	108	113	150	169	184	207	216	220	242	266	294	295	297	305	311	338	359
ℓ243	0	84	102	109	114	151	170	185	208	217	221	243	267	295	296	298	306	312	339
ℓ244	1	85	103	110	115	152	171	186	209	218	222	244	268	296	297	299	307	313	340
ℓ245	2	86	104	111	116	153	172	187	210	219	223	245	269	297	298	300	308	314	341
ℓ246	3	87	105	112	117	154	173	188	211	220	224	246	270	298	299	301	309	315	342
ℓ247	4	88	106	113	118	155	174	189	212	221	225	247	271	299	300	302	310	316	343
ℓ248	5	89	107	114	119	156	175	190	213	222	226	248	272	300	301	303	311	317	344
ℓ249	6	90	108	115	120	157	176	191	214	223	227	249	273	301	302	304	312	318	345
ℓ250	7	91	109	116	121	158	177	192	215	224	228	250	274	302	303	305	313	319	346
ℓ251	8	92	110	117	122	159	178	193	216	225	229	251	275	303	304	306	314	320	347
ℓ252	9	93	111	118	123	160	179	194	217	226	230	252	276	304	305	307	315	321	348
ℓ253	10	94	112	119	124	161	180	195	218	227	231	253	277	305	306	308	316	322	349
ℓ254	11	95	113	120	125	162	181	196	219	228	232	254	278	306	307	309	317	323	350
ℓ255	12	96	114	121	126	163	182	197	220	229	233	255	279	307	308	310	318	324	351
ℓ256	13	97	115	122	127	164	183	198	221	230	234	256	280	308	309	311	319	325	352
ℓ257	14	98	116	123	128	165	184	199	222	231	235	257	281	309	310	312	320	326	353
ℓ258	15	99	117	124	129	166	185	200	223	232	236	258	282	310	311	313	321	327	354
ℓ259	16	100	118	125	130	167	186	201	224	233	237	259	283	311	312	314	322	328	355
ℓ260	17	101	119	126	131	168	187	202	225	234	238	260	284	312	313	315	323	329	356
ℓ261	18	102	120	127	132	169	188	203	226	235	239	261	285	313	314	316	324	330	357
ℓ262	19	103	121	128	133	170	189	204	227	236	240	262	286	314	315	317	325	331	358
ℓ263	20	104	122	129	134	171	190	205	228	237	241	263	287	315	316	318	326	332	359
ℓ264	0	21	105	123	130	135	172	191	206	229	238	242	264	288	316	317	319	327	333
ℓ265	1	22	106	124	131	136	173	192	207	230	239	243	265	289	317	318	320	328	334
ℓ266	2	23	107	125	132	137	174	193	208	231	240	244	266	290	318	319	321	329	335
ℓ267	3	24	108	126	133	138	175	194	209	232	241	245	267	291	319	320	322	330	336
ℓ268	4	25	109	127	134	139	176	195	210	233	242	246	268	292	320	321	323	331	337
ℓ269	5	26	110	128	135	140	177	196	211	234	243	247	269	293	321	322	324	332	338
ℓ270	6	27	111	129	136	141	178	197	212	235	244	248	270	294	322	323	325	333	339
ℓ271	7	28	112	130	137	142	179	198	213	236	245	249	271	295	323	324	326	334	340
ℓ272	8	29	113	131	138	143	180	199	214	237	246	250	272	296	324	325	327	335	341
ℓ273	9	30	114	132	139	144	181	200	215	238	247	251	273	297	325	326	328	336	342
ℓ274	10	31	115	133	140	145	182	201	216	239	248	252	274	298	326	327	329	337	343
ℓ275	11	32	116	134	141	146	183	202	217	240	249	253	275	299	327	328	330	338	344
ℓ276	12	33	117	135	142	147	184	203	218	241	250	254	276	300	328	329	331	339	345
ℓ277	13	34	118	136	143	148	185	204	219	242	251	255	277	301	329	330	332	340	346
ℓ278	14	35	119	137	144	149	186	205	220	243	252	256	278	302	330	331	333	341	347
ℓ279	15	36	120	138	145	150	187	206	221	244	253	257	279	303	331	332	334	342	348
ℓ280	16	37	121	139	146	151	188	207	222	245	254	258	280	304	332	333	335	343	349
ℓ281	17	38	122	140	147	152	189	208	223	246	255	259	281	305	333	334	336	344	350
ℓ282	18	39	123	141	148	153	190	209	224	247	256	260	282	306	334	335	337	345	351
ℓ283	19	40	124	142	149	154	191	210	225	248	257	261	283	307	335	336	338	346	352
ℓ284	20	41	125	143	150	155	192	211	226	249	258	262	284	308	336	337	339	347	353
ℓ285	21	42	126	144	151	156	193	212	227	250	259	263	285	309	337	338	340	348	354
ℓ286	22	43	127	145	152	157	194	213	228	251	260	264	286	310	338	339	341	349	355
ℓ287	23	44	128	146	153	158	195	214	229	252	261	265	287	311	339	340	342	350	356
ℓ288	24	45	129	147	154	159	196	215	230	253	262	266	288	312	340	341	343	351	357
ℓ289	25	46	130	148	155	160	197	216	231	254	263	267	289	313	341	342	344	352	358
ℓ290	26	47	131	149	156	161	198	217	232	255	264	268	290	314	342	343	345	353	359
ℓ291	0	27	48	132	150	157	162	199	218	233	256	265	269	291	315	343	344	346	354
ℓ292	1	28	49	133	151	158	163	200	219	234	257	266	270	292	316	344	345	347	355
ℓ293	2	29	50	134	152	159	164	201	220	235	258	267	271	293	317	345	346	348	356
ℓ294	3	30	51	135	153	160	165	202	221	236	259	268	272	294	318	346	347	349	357
ℓ295	4	31	52	136	154	161	166	203	222	237	260	269	273	295	319	347	348	350	358
ℓ296	5	32	53	137	155	162	167	204	223	238	261	270	274	296	320	348	349	351	359
ℓ297	0	6	33	54	138	156	163	168	205	224	239	262	271	275	297	321	349	350	352
ℓ298	1	7	34	55	139	157	164	169	206	225	240	263	272	276	298	322	350	351	353
ℓ299	2	8	35	56	140	158	165	170	207	226	241	264	273	277	299	323	351	352	354
ℓ300	3	9	36	57	141	159	166	171	208	227	242	265	274	278	300	324	352	353	355
ℓ301	4	10	37	58	142	160	167	172	209	228	243	266	275	279	301	325	353	354	356
ℓ302	5	11	38	59	143	161	168	173	210	229	244	267	276	280	302	326	354	355	357
ℓ303	6	12	39	60	144	162	169	174	211	230	245	268	277	281	303	327	355	356	358
ℓ304	7	13	40	61	145	163	170	175	212	231	246	269	278	282	304	328	356	357	359
ℓ305	0	8	14	41	62	146	164	171	176	213	232	247	270	279	283	305	329	357	358
ℓ306	1	9	15	42	63	147	165	172	177	214	233	248	271	280	284	306	330	358	359
ℓ307	0	2	10	16	43	64	148	166	173	178	215	234	249	272	281	285	307	331	359
ℓ308	0	1	3	11	17	44	65	149	167	174	179	216	235	250	273	282	286	308	332
ℓ309	1	2	4	12	18	45	66	150	168	175	180	217	236	251	274	283	287	309	333
ℓ310	2	3	5	13	19	46	67	151	169	176	181	218	237	252	275	284	288	310	334
ℓ311	3	4	6	14	20	47	68	152	170	177	182	219	238	253	276	285	289	311	335
ℓ312	4	5	7	15	21	48	69	153	171	178	183	220	239	254	277	286	290	312	336
ℓ313	5	6	8	16	22	49	70	154	172	179	184	221	240	255	278	287	291	313	337
ℓ314	6	7	9	17	23	50	71	155	173	180	185	222	241	256	279	288	292	314	338
ℓ315	7	8	10	18	24	51	72	156	174	181	186	223	242	257	280	289	293	315	339
ℓ316	8	9	11	19	25	52	73	157	175	182	187	224	243	258	281	290	294	316	340
ℓ317	9	10	12	20	26	53	74	158	176	183	188	225	244	259	282	291	295	317	341
ℓ318	10	11	13	21	27	54	75	159	177	184	189	226	245	260	283	292	296	318	342
ℓ319	11	12	14	22	28	55	76	160	178	185	190	227	246	261	284	293	297	319	343
ℓ320	12	13	15	23	29	56	77	161	179	186	191	228	247	262	285	294	298	320	344
ℓ321	13	14	16	24	30	57	78	162	180	187	192	229	248	263	286	295	299	321	345
ℓ322	14	15	17	25	31	58	79	163	181	188	193	230	249	264	287	296	300	322	346
ℓ323	15	16	18	26	32	59	80	164	182	189	194	231	250	265	288	297	301	323	347
ℓ324	16	17	19	27	33	60	81	165	183	190	195	232	251	266	289	298	302	324	348
ℓ325	17	18	20	28	34	61	82	166	184	191	196	233	252	267	290	299	303	325	349
ℓ326	18	19	21	29	35	62	83	167	185	192	197	234	253	268	291	300	304	326	350
ℓ327	19	20	22	30	36	63	84	168	186	193	198	235	254	269	292	301	305	327	351
ℓ328	20	21	23	31	37	64	85	169	187	194	199	236	255	270	293	302	306	328	352
ℓ329	21	22	24	32	38	65	86	170	188	195	200	237	256	271	294	303	307	329	353
ℓ330	22	23	25	33	39	66	87	171	189	196	201	238	257	272	295	304	308	330	354
ℓ331	23	24	26	34	40	67	88	172	190	197	202	239	258	273	296	305	309	331	355
ℓ332	24	25	27	35	41	68	89	173	191	198	203	240	259	274	297	306	310	332	356
ℓ333	25	26	28	36	42	69	90	174	192	199	204	241	260	275	298	307	311	333	357
ℓ334	26	27	29	37	43	70	91	175	193	200	205	242	261	276	299	308	312	334	358
ℓ335	27	28	30	38	44	71	92	176	194	201	206	243	262	277	300	309	313	335	359
ℓ336	0	28	29	31	39	45	72	93	177	195	202	207	244	263	278	301	310	314	336
ℓ337	1	29	30	32	40	46	73	94	178	196	203	208	245	264	279	302	311	315	337
ℓ338	2	30	31	33	41	47	74	95	179	197	204	209	246	265	280	303	312	316	338
ℓ339	3	31	32	34	42	48	75	96	180	198	205	210	247	266	281	304	313	317	339
ℓ340	4	32	33	35	43	49	76	97	181	199	206	211	248	267	282	305	314	318	340
ℓ341	5	33	34	36	44	50	77	98	182	200	207	212	249	268	283	306	315	319	341
ℓ342	6	34	35	37	45	51	78	99	183	201	208	213	250	269	284	307	316	320	342
ℓ343	7	35	36	38	46	52	79	100	184	202	209	214	251	270	285	308	317	321	343
ℓ344	8	36	37	39	47	53	80	101	185	203	210	215	252	271	286	309	318	322	344
ℓ345	9	37	38	40	48	54	81	102	186	204	211	216	253	272	287	310	319	323	345
ℓ346	10	38	39	41	49	55	82	103	187	205	212	217	254	273	288	311	320	324	346
ℓ347	11	39	40	42	50	56	83	104	188	206	213	218	255	274	289	312	321	325	347
ℓ348	12	40	41	43	51	57	84	105	189	207	214	219	256	275	290	313	322	326	348
ℓ349	13	41	42	44	52	58	85	106	190	208	215	220	257	276	291	314	323	327	349
ℓ350	14	42	43	45	53	59	86	107	191	209	216	221	258	277	292	315	324	328	350
ℓ351	15	43	44	46	54	60	87	108	192	210	217	222	259	278	293	316	325	329	351
ℓ352	16	44	45	47	55	61	88	109	193	211	218	223	260	279	294	317	326	330	352
ℓ353	17	45	46	48	56	62	89	110	194	212	219	224	261	280	295	318	327	331	353
ℓ354	18	46	47	49	57	63	90	111	195	213	220	225	262	281	296	319	328	332	354
ℓ355	19	47	48	50	58	64	91	112	196	214	221	226	263	282	297	320	329	333	355
ℓ356	20	48	49	51	59	65	92	113	197	215	222	227	264	283	298	321	330	334	356
ℓ357	21	49	50	52	60	66	93	114	198	216	223	228	265	284	299	322	331	335	357
ℓ358	22	50	51	53	61	67	94	115	199	217	224	229	266	285	300	323	332	336	358
ℓ359	23	51	52	54	62	68	95	116	200	218	225	230	267	286	301	324	333	337	359

.

The lines containing the point $\infty$ of the plane are found by tacking the cosets of the unique subgroup of order 18 of ${\mathbb{Z}}_{360}$ written in

Table A3. The lines containing the point $\infty$ of AG(2,19).

m0	$\infty$	0	20	40	60	80	100	120	140	160	180	200	220	240	260	280	300	320	340
m1	$\infty$	1	21	41	61	81	101	121	141	161	181	201	221	241	261	281	301	321	341
m2	$\infty$	2	22	42	62	82	102	122	142	162	182	202	222	242	262	282	302	322	342
m3	$\infty$	3	23	43	63	83	103	123	143	163	183	203	223	243	263	283	303	323	343
m4	$\infty$	4	24	44	64	84	104	124	144	164	184	204	224	244	264	284	304	324	344
m5	$\infty$	5	25	45	65	85	105	125	145	165	185	205	225	245	265	285	305	325	345
m6	$\infty$	6	26	46	66	86	106	126	146	166	186	206	226	246	266	286	306	326	346
m7	$\infty$	7	27	47	67	87	107	127	147	167	187	207	227	247	267	287	307	327	347
m8	$\infty$	8	28	48	68	88	108	128	148	168	188	208	228	248	268	288	308	328	348
m9	$\infty$	9	29	49	69	89	109	129	149	169	189	209	229	249	269	289	309	329	349
m10	$\infty$	10	30	50	70	90	110	130	150	170	190	210	230	250	270	290	310	330	350
m11	$\infty$	11	31	51	71	91	111	131	151	171	191	211	231	251	271	291	311	331	351
m12	$\infty$	12	32	52	72	92	112	132	152	172	192	212	232	252	272	292	312	332	352
m13	$\infty$	13	33	53	73	93	113	133	153	173	193	213	233	253	273	293	313	333	353
m14	$\infty$	14	34	54	74	94	114	134	154	174	194	214	234	254	274	294	314	334	354
m15	$\infty$	15	35	55	75	95	115	135	155	175	195	215	235	255	275	295	315	335	355
m16	$\infty$	16	36	56	76	96	116	136	156	176	196	216	236	256	276	296	316	336	356
m17	$\infty$	17	37	57	77	97	117	137	157	177	197	217	237	257	277	297	317	337	357
m18	$\infty$	18	38	58	78	98	118	138	158	178	198	218	238	258	278	298	318	338	358
m19	$\infty$	19	39	59	79	99	119	139	159	179	199	219	239	259	279	299	319	339	359

.

The orbits of the points, different from $\infty$, under the unique subgroup of order 20 of ${\mathbb{Z}}_{360}$ define a partition in 18 conics of the set of points of AG(2,19)−$\left\{\infty \right\}$ written in

Table A4. The partition of AG(2,19)−$\left\{\infty \right\}$ in conics.

c0	0	18	36	54	72	90	108	126	144	162	180	198	216	234	252	270	288	306	324	342
c1	1	19	37	55	73	91	109	127	145	163	181	199	217	235	253	271	289	307	325	343
c2	2	20	38	56	74	92	110	128	146	164	182	200	218	236	254	272	290	308	326	344
c3	3	21	39	57	75	93	111	129	147	165	183	201	219	237	255	273	291	309	327	345
c4	4	22	40	58	76	94	112	130	148	166	184	202	220	238	256	274	292	310	328	346
c5	5	23	41	59	77	95	113	131	149	167	185	203	221	239	257	275	293	311	329	347
c6	6	24	42	60	78	96	114	132	150	168	186	204	222	240	258	276	294	312	330	348
c7	7	25	43	61	79	97	115	133	151	169	187	205	223	241	259	277	295	313	331	349
c8	8	26	44	62	80	98	116	134	152	170	188	206	224	242	260	278	296	314	332	350
c9	9	27	45	63	81	99	117	135	153	171	189	207	225	243	261	279	297	315	333	351
c10	10	28	46	64	82	100	118	136	154	172	190	208	226	244	262	280	298	316	334	352
c11	11	29	47	65	83	101	119	137	155	173	191	209	227	245	263	281	299	317	335	353
c12	12	30	48	66	84	102	120	138	156	174	192	210	228	246	264	282	300	318	336	354
c13	13	31	49	67	85	103	121	139	157	175	193	211	229	247	265	283	301	319	337	355
c14	14	32	50	68	86	104	122	140	158	176	194	212	230	248	266	284	302	320	338	356
c15	15	33	51	69	87	105	123	141	159	177	195	213	231	249	267	285	303	321	339	357
c16	16	34	52	70	88	106	124	142	160	178	196	214	232	250	268	286	304	322	340	358
c17	17	35	53	71	89	107	125	143	161	179	197	215	233	251	269	287	305	323	341	359

.

The orbits of the points, different from $\infty$, under the unique subgroup of order 4 of ${\mathbb{Z}}_{360}$ define a partition in 4-arcs written in

Table A5. The partition of AG(2,19)−$\left\{\infty \right\}$ in 4-arcs.

C0	0	90	180	270
C1	1	91	181	271
C2	2	92	182	272
C3	3	93	183	273
C4	4	94	184	274
C5	5	95	185	275
C6	6	96	186	276
C7	7	97	187	277
C8	8	98	188	278
C9	9	99	189	279
C10	10	100	190	280
C11	11	101	191	281
C12	12	102	192	282
C13	13	103	193	283
C14	14	104	194	284
C15	15	105	195	285
C16	16	106	196	286
C17	17	107	197	287
C18	18	108	198	288
C19	19	109	199	289
C20	20	110	200	290
C21	21	111	201	291
C22	22	112	202	292
C23	23	113	203	293
C24	24	114	204	294
C25	25	115	205	295
C26	26	116	206	296
C27	27	117	207	297
C28	28	118	208	298
C29	29	119	209	299
C30	30	120	210	300
C31	31	121	211	301
C32	32	122	212	302
C33	33	123	213	303
C34	34	124	214	304
C35	35	125	215	305
C36	36	126	216	306
C37	37	127	217	307
C38	38	128	218	308
C39	39	129	219	309
C40	40	130	220	310
C41	41	131	221	311
C42	42	132	222	312
C43	43	133	223	313
C44	44	134	224	314
C45	45	135	225	315
C46	46	136	226	316
C47	47	137	227	317
C48	48	138	228	318
C49	49	139	229	319
C50	50	140	230	320
C51	51	141	231	321
C52	52	142	232	322
C53	53	143	233	323
C54	54	144	234	324
C55	55	145	235	325
C56	56	146	236	326
C57	57	147	237	327
C58	58	148	238	328
C59	59	149	239	329
C60	60	150	240	330
C61	61	151	241	331
C62	62	152	242	332
C63	63	153	243	333
C64	64	154	244	334
C65	65	155	245	335
C66	66	156	246	336
C67	67	157	247	337
C68	68	158	248	338
C69	69	159	249	339
C70	70	160	250	340
C71	71	161	251	341
C72	72	162	252	342
C73	73	163	253	343
C74	74	164	254	344
C75	75	165	255	345
C76	76	166	256	346
C77	77	167	257	347
C78	78	168	258	348
C79	79	169	259	349
C80	80	170	260	350
C81	81	171	261	351
C82	82	172	262	352
C83	83	173	263	353
C84	84	174	264	354
C85	85	175	265	355
C86	86	176	266	356
C87	87	177	267	357
C88	88	178	268	358
C89	89	179	269	359

.

Let us take into account C₀. The first quadrilateral C_i, i = 1, 2,…, 11, such that C₀ and C_i are simultaneously inscribed and circumscribed, is C₉. Thus, the union C₀∪C₉ is the classical representation of the Möbius–Kantor configuration (8₃), cf. [20]. Finally, we prove the cyclic partition of AG(2,19). Since ${\sigma }^i\left(C_0\right)=C_i$, we have that

$$\begin{gathered} {\sigma }^0\left(C_0\cup C_9\right)=C_0\cup C_9; {\sigma }^2\left(C_0\cup C_9\right)=C_2\cup C_{11}; {\sigma }^4\left(C_0\cup C_9\right)={\sigma }^2\left(C_2\cup C_{11}\right)= \\ {C}_4\cup C_{13}; {\sigma }^6\left(C_0\cup C_9\right)={\sigma }^2\left(C_4\cup C_{13}\right)=C_6\cup C_{15}; {{\sigma }^8\left(C_0\cup C_9\right)=\sigma }^2\left(C_6\cup C_{15}\right)= \\ {C}_8\cup C_{17}; {\sigma }^{10}\left(C_0\cup C_9\right){=\sigma }^2\left(C_8\cup C_{17}\right)=C_{10}\cup C_{19}; {\sigma }^{12}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{10}\cup C_{19}\right)=C_{12}\cup C_{21}; \\ {\sigma }^{14}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{12}\cup C_{21}\right)=C_{14}\cup C_{23}; {\sigma }^{16}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{14}\cup C_{23}\right)=C_{16}\cup C_{25}; \\ {\sigma }^{18}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{16}\cup C_{25}\right)=C_{18}\cup C_{27}; {\sigma }^{20}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{18}\cup C_{27}\right)=C_{20}\cup C_{29}; \\ {\sigma }^{22}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{20}\cup C_{29}\right)=C_{22}\cup C_{31}; {\sigma }^{24}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{22}\cup C_{31}\right)=C_{24}\cup C_{33}; \\ {\sigma }^{26}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{24}\cup C_{33}\right)=C_{26}\cup C_{35}; {\sigma }^{28}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{26}\cup C_{35}\right)=C_{28}\cup C_{37}; \\ {\sigma }^{30}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{28}\cup C_{37}\right)=C_{30}\cup C_{39}; {\sigma }^{32}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{30}\cup C_{39}\right)=C_{32}\cup C_{41}; \\ {\sigma }^{34}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{32}\cup C_{41}\right)=C_{34}\cup C_{43}; {\sigma }^{36}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{34}\cup C_{43}\right)=C_{36}\cup C_{45}; \\ {\sigma }^{38}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{36}\cup C_{45}\right)=C_{38}\cup C_{47}; {\sigma }^{40}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{38}\cup C_{47}\right)=C_{40}\cup C_{49}; \\ {\sigma }^{42}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{40}\cup C_{49}\right)=C_{42}\cup C_{51}; {\sigma }^{44}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{42}\cup C_{51}\right)=C_{44}\cup C_{53}; \\ {\sigma }^{46}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{44}\cup C_{53}\right)=C_{46}\cup C_{55}; {\sigma }^{48}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{46}\cup C_{55}\right)=C_{48}\cup C_{57}; \\ {\sigma }^{50}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{48}\cup C_{57}\right)=C_{50}\cup C_{59}; {\sigma }^{52}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{50}\cup C_{59}\right)=C_{52}\cup C_{61}; \\ {\sigma }^{54}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{52}\cup C_{61}\right)=C_{54}\cup C_{63}; {\sigma }^{56}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{54}\cup C_{63}\right)=C_{56}\cup C_{65}; \\ {\sigma }^{58}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{56}\cup C_{65}\right)=C_{58}\cup C_{67}; {\sigma }^{60}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{58}\cup C_{67}\right)=C_{60}\cup C_{69}; \\ {\sigma }^{62}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{60}\cup C_{69}\right)=C_{62}\cup C_{71}; {\sigma }^{64}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{62}\cup C_{71}\right)=C_{64}\cup C_{73}; \\ {\sigma }^{66}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{64}\cup C_{73}\right)=C_{66}\cup C_{75}; {\sigma }^{68}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{66}\cup C_{75}\right)=C_{68}\cup C_{77}; \\ {\sigma }^{70}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{68}\cup C_{77}\right)=C_{70}\cup C_{79}; {\sigma }^{72}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{70}\cup C_{79}\right)=C_{72}\cup C_{81}; \\ {\sigma }^{74}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{72}\cup C_{81}\right)=C_{74}\cup C_{83}; {\sigma }^{76}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{74}\cup C_{83}\right)=C_{76}\cup C_{85}; \\ {\sigma }^{78}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{76}\cup C_{85}\right)=C_{78}\cup C_{87}; {\sigma }^{80}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{78}\cup C_{87}\right)=C_{80}\cup C_{89}; \\ {\sigma }^{82}\left(C_0\cup C_9\right){=\sigma }^2\left(C_{80}\cup C_{89}\right)=C_1\cup C_{82}; {\sigma }^{84}\left(C_0\cup C_9\right){=\sigma }^2\left(C_1\cup C_{82}\right)=C_3\cup C_{84}; \\ {\sigma }^{86}\left(C_0\cup C_9\right){=\sigma }^2\left(C_3\cup C_{84}\right)=C_5\cup C_{86}; {\sigma }^{88}\left(C_0\cup C_9\right){=\sigma }^2\left(C_5\cup C_{86}\right)=C_7\cup C_{88}. \end{gathered}$$

Therefore, we obtain the cyclic partition of AG(2,19). □

6. Discussion

This work presents a comprehensive study of the problem, initially proposed in [19], of partitioning the affine plane of order q, q≡7 mod 12, into one point ∞ and ${\displaystyle \frac{q^2-1}{8}}$ Möbius–Kantor configurations 8₃. Focusing on the cyclic structure, we establish a complete mathematical construction. We provide two concrete examples demonstrating its effectiveness. For further research, we consider the order q, q≡1 mod 12, in which the orbits of the points, different from $\infty$, under the unique subgroup of order 4 of ${\mathbb{Z}}_{q^2-1}$ do not define a 4-arc partition, but a 4 collinear points partition, and, therefore, the proposed construction does not work and other ideas are needed.