西 南 交 通 大 学 学 报
第 54 卷 第 6 期
2019 年 12 月
JOURNAL OF SOUTHWEST JIAOTONG UNIVERSITY
Vol. 54 No. 6
Dec. 2019
ISSN: 0258-2724 DOI：10.35741/issn.0258-2724.54.6.46
Research Article Mathematics
C
UBIC
A
RCS IN THE
P
ROJECTIVE
P
LANE
O
VER A
F
INITE
F
IELD
OF
O
RDER
T
WENTY
T
HREE
二十三阶有限域上射影平面上的立体弧
Najm A.M. Al-Seraji, Asraa A. Monshed
Department of Mathematics, College of Science, University of Mustansiriyah
P.O. Box: 14022, Palestine St., Baghdad, Iraq, [email protected],[email protected]
Abstract
In this research we are interested in finding all the different cubic curves over a finite projective plane of order twenty-three, learning which of them is complete or not, constructing the stabilizer groups of the cubics in , studying the properties of these groups, and, finally, introducing the relation between the subject of coding theory and the projective plane of order twenty three.
Keywords:Stabilizer Group, Arc, Cubic Curve, Coding Theory
摘要 在这项研究中，我们有兴趣在23个阶的有限射影平面上找到所有不同的三次曲线，了解它
们中的哪一个完整或不完整，构造其中的三次稳定子组，研究这些组的性质，以及最后，介绍 了编码理论的主题与二十三级射影平面之间的关系。
关键词: 稳定器组，弧，三次曲线，编码理论
I.
I
NTRODUCTIONThe main goal of this research is to find all the
distance cubic curves of and
construct the stabilizer group with finding the linear transformations groups in
which, in its element, are considered the
non-singular matrices in
for some and satisfying for all t in and
be any arcs, also we find the maximal size of a complete arc of degree three can be constructed from each incomplete arc. We also find the relation between a topic of coding theory and the projective plane of order twenty three.
The first step is finding the points and lines of which contains of 553 point and line, with 24 point on every line and 24 line on each point.
The second step is finding the number of rational inflexion and rational points, we denote
be the number of distance cubic curves with exactly rational inflexions, and denote be the total number, where
.
The third step is classifying the non-singular cubic curves to non-singular one rational inflexion, no rational inflexion, three rational inflexions and study each one of them
in .
The forth step is to construct the stabilizer groups of the cubic in and study the properties of these groups.
A code with parameters has the following good properties if:
1. The parameter which length of a code is small.
2. The parameter which dimension of a code is large.
3. The parameter which minimum distance of a code C is large.
The relation between a topic of coding theory and the projective geometry is:
( in exists if and only if
a code exists.
Finally, the end of this research is devoted to give a summary of this work.
For more details and information see [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12].
The next step is introducing some definitions which are used through the research.
A. Definition 1.1 [1]
A rational inflexion point P of a non-singular cubic curve is one for which the unique tangent at P has three – points contact; that is, the unique tangent at an inflexion P of has no other point in common with the curve .
B. Definition 1.2 [1]
An arc K of degree n denoted by in
, is a set of k point no n + 1 of them are collinear but some n collinear.
C. Definition 1.3 [1]
A is considered complete if it is not contained with
II.
T
HEC
LASSIFICATION OFC
UBICA
RCS INPut ,
where is the primitive polynomial over .
This means ( is irreducible over and reducible over ) with as the primitive element of , has three zeros
and in .
, has 533 points and lines, such that the number of points on each line is 24 and the number of lines passing through each point is 24.
The companion matrix of is
C( . The points of PG (2,23) are where .
The lines of PG(2,23) are
where be the line passing through points with z=0, the numeral form of is: 0, 1, 8, 19, 50, 59, 159, 164, 196, 226, 232, 260, 314, 334, 349, 412, 425, 429, 441, 468, 482, 506, 528, 551}.
The equations of lines in are: .
III.
N
UMBER OFN
ON-S
INGULARP
LANEC
UBICS ANDT
HEIRR
ATIONALP
OINTSLet be the total number of projectively inequivalent cubics. Let for be the number of projectively inequivalent cubics with exactly rational inflexions. Hence,
.
A. Theorem 3.1 [1]
Here, both certain Legendre and Legendre-Jacobi symbols are used:
B. Corollary 3.2
Over , the total number of projectivity inequivalent cubics is 68; that is .
C. Theorem 3.3 [1]
Over , the number of rational inflexions of non-singular plane cubics is zero, one, three, or nine.
D. Corollary 3.4
Over the number of rational inflexions of non – singular plane cubic is zero, one, and three
IV.
N
UMBER OFR
ATIONALP
OINTS ON AC
UBICC
URVELet denote the maximum number of rational points on any non-singular cubic curve over and the minimum number.
A. Theorem 4.1 [1]
The number takes every value between
and if and only if
(i) or (ii) with
.
B. Corollary 4.2
Since in where
.
V.
N
ON-S
INGULARP
LANEC
UBICS WITHN
OR
ATIONALI
NFLEXIONWe introduce the following lemma to construct the cubic curves with no rational inflexion:
A. Lemma 5.1 [1]
Over , a non-singular
cubic with no rational inflexions has canonical form = where is irreducible. B. Corollary 5.2 In , the polynomial is irreducible when . C. Lemma 5.3 [1]
The curve is equiaharmonic for , harmonic
.
D. Corollary 5.4
In , the curve in Lemma 5.1 is equianharmoic for c = - 8, 0, 11, harmonic for c = - 1, 9.
If we substitute the points of PG(2, 23) in Lemma 5.3, we get:
The number of the cubic curves of size 15 is 8.
The number of the cubic curves of size 18 is 32.
The number of the cubic curves of size 21 is 24.
The number of the cubic curves of size 24 is 48.
The number of the cubic curves of size 27 is 24.
The number of the cubic curves of size 30 is 32.
The number of the cubic curves of size 33 is 8.
After finding the transformations between previous cubics, Table 1 shows that the canonical form for the non-singular plane cubic curves with exactly no rational inflexion; type of a non-singular cubic curve of , where are respectively the general, equianharmonic types; number of rational points; description is complete or incomplete; and the maximal size of a complete arc contains each cubic curve.
Table 1.
Non-singular plane cubic curves with no rational inflexion
Max. arc Des. Size Type Canonical form No. 25 Inc. 18 G 1
29 Inc. 21 G 2 37 Inc. 30 G 3 31 Inc. 27 G 4 31 Inc. 24 G 5 31 Inc. 21 E 6 27 Inc. 21 G 7 22 Inc. 15 G 8 41 Inc. 33 G 9 35 Inc. 30 H 10 26 Inc. 18 G 11 30 Inc. 24 G 12 37 Inc. 30 G 13 35 Inc. 27 E 14 29 Inc. 24 E 15 31 Inc. 24 H 16 31 Inc. 24 G 17 38 Inc. 30 G 18 34 Inc. 27 G 19 23 Inc. 18 G 20 25 Inc. 18 G 21 25 Inc. 24 E 22 E. Remark 5.5
The stabilizer groups of all cubic curves in Table 1 are .
VI.
N
ON-S
INGULARP
LANEC
UBICC
URVES WITHO
NER
ATIONALI
NFLEXIONWe introduce the following lemma to construct cubic curves with one rational inflexion:
A. Lemma 6.1 [1]
A non-singular plane cubic curve defined over with at least one inflexion. The curve has the canonical form
,
where .
B. Corollary 6.2
1. There are 440 ordered pairs (c, d) that
satisfy the equation
2. For (c, d) =
.
C. Remark 6.3 [1]
The curve in Lemma 6.1 is as follows:
The cubic curves of size 17
After finding transformations between previous cubics, Table 2 shows that the canonical form for the non-singular plane cubic; curves with exactly one rational inflexion; type of a non-singular cubic curve of , where are
respectively the general equiharmonic types; number of rational points; description is complete
or incomplete; and maximal size of a complete arc contains each curve.
Table 2.
Non-singular plane cubic curves with one rational inflexion
Max. arc De. Size Type Canonical form No. 32 Inc. 26 G 1 31 Inc. 22 G 2 32 Inc. 25 G 3 28 Inc. 22 G 4 26 Inc. 19 G 5 23 Inc. 17 G 6 24 Inc. 16 G 7 38 Inc. 32 G 8 28 Inc. 20 G 9 32 Inc. 23 G 10 36 Inc. 28 G 11 40 Inc. 32 G 12 30 Inc. 23 G 13 34 Inc. 28 G 14 28 Inc. 28 G 15 33 Inc. 29 G 16 35 Inc. 20 G 17 28 Inc. 20 G 18 26 Inc. 28 G 19 36 Inc. 28 G 20 32 Inc. 25 G 21 39 Inc. 31 G 22 21 Inc. 16 G 23 26 Com. 26 G 24 D. Remark 6.4
The stabilizer groups of all cubic curves in Table 2 are .
VII.
N
ON-S
INGULARP
LANEC
UBICS WITHT
HREER
ATIONALI
NFLEXIONSWe introduce the following lemma to construct cubic curves with three rational inflexions:
A. Lemma 7.1 [1]
There are distinct projective plane cubic curves with three collinear rational inflexions, such that the inflexional tangents are concurrent.
The canonical forms are as follows:
w .
B. Corollary 7.2
In , the polynomial has three rational inflexions.
C. Theorem 7.3 [1]
A non-singular plane cubic curve over three collinear rational inflexions and non-concurrent inflexional tangents has three rational inflexions and a canonical form
where . D. Corollary 7.4 . E. Lemma 7.5 [1] The polynomial is as follows: 1) singular and irreducible if
3) harmonic if
F. Corollary 7.6
In the polynomial
is: 1) singular and irreducible if e = - 6, 2) equianharmonic if 3) of no harmonic type of .
Table 3 is showing that the canonical form for the non-singular plane cubic curves with exactly three rational inflexions; type of a non- singular cubic curve of , where are respectively the general equianharmonic types when the inflexional tangents are not concurrent, and type when they are concurrent; number of rational points; description is complete or incomplete; and the maximal size of a complete arc contains each curve.
Table 3.
Non-singular plane cubic curves with three rational inflexions
Max. arc Des. Size Type Canonical form No. 25 Inc. 24 1 31 Inc. 24 G 2 34 Inc. 24 E 3 25 Inc. 18 G 4 41 Inc. 33 G 5 37 Inc. 30 G 6 29 Inc. 21 G 7 32 Inc. 24 G 8 22 Inc. 15 G 9 25 Inc. 18 G 10 37 Inc. 24 G 11 39 Inc. 30 G 12 29 Inc. 21 G 13 32 Inc. 24 G 14 35 Inc. 27 G 15 30 Inc. 21 G 16 29 Inc. 18 G 17 38 Inc. 30 G 18 37 Inc. 27 G 19 25 Inc. 18 G 20 32 Inc. 27 G 21 36 Inc. 30 G 22 G. Remark 7.7
The stabilizer groups of all cubic curves in Table 3 are .
VIII.
T
HEA
PPLICATIONS OF THEP
ROJECTIVEP
LANE OFO
RDERT
WENTYT
HREE INC
ODINGT
HEORYTables 4, 6, 8 are the found parameters of PG(2, 23) for any cubic.
Table 4.
The parameters and e on PG(2, 23) with no inflexion No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 No. e 1 7 [18, 3, 15] 2 8 [21, 3, 18] 3 13 [30, 3, 27]
4 11 [27, 3, 24] 5 10 [24, 3, 21] 6 8 [21, 3, 18] 7 8 [21, 3, 18] 8 5 [15, 3, 12] 9 14 [33, 3, 30] 10 13 [30, 3, 27] 11 7 [18, 3, 15] 12 10 [24, 3, 21] 13 13 [30, 3, 27] 14 11 [27, 3, 24] 15 10 [24, 3, 21] 16 10 [24, 3, 21] 17 10 [24, 3, 21] 18 13 [30, 3, 27] 19 11 [27, 3, 24] 20 7 [18, 3, 15] 21 7 [18, 3, 15] 22 10 [24, 3, 21]
Take the cubic curve:
in
Table 4. The points of PG(2, 23) on are given in Table 5, as follows:
Table 5.
The points of PG(2, 23) on
The generator matrix G of
= is:
G =
Such that x = for all x
in with
represents rows of G and
Take the cubic
curve in Table
6.
Table 6.
The parameters and e on PG(2, 23) with one inflexion
e [ Canonical form No. 11 [26, 3, 23] 1 9 [22, 3, 19] 2 10 [25, 3, 22] 3 9 [22, 3, 19] 4 7 [19, 3, 16] 5 6 [17, 3, 14] 6 6 [16, 3, 13] 7 14 [32, 3, 29] 8 8 [20, 3, 17] 9 9 [23, 3, 20] 10 12 [28, 3, 25] 11 14 [32, 3, 29] 12 9 [23, 3, 20] 13 12 [28, 3, 25] 14
8 [20, 3, 17] 15 12 [28, 3, 25] 16 12 [29, 3, 26] 17 8 [20, 3, 17] 18 8 [20, 3, 17] 19 12 [28, 3, 25] 20 10 [25, 3, 22] 21 13 [31, 3, 28] 22 6 [16, 3, 13] 23 11 [26, 3, 23] 24
The points of PG(2, 23) on are given in Table 7 as:
Table 7.
The points of PG(2, 23) on
The generator matrix H of
= is
H =
Such that y = for all y in
= with
represents rows of H and Let, D = = . Since D([0, 0, 1]) = 0. Therefore, [0, 0, 1] is an inflexion point of . Table 8.
The parameters and e on PG(2, 23) with three inflexions
e [ Canonical form No. 10 [24, 3, 21] 1 10 [24, 3, 21] 2 10 [24, 3, 21] 3 7 [18, 3, 15] 4 14 [33, 3, 30] 5 13 [30, 3, 27] 6 8 [21, 3, 18] 7 10 [24, 3, 21] 8 5 [15, 3, 12] 9 7 [18, 3, 15] 10 10 [24, 3, 21] 11 13 [30, 3, 27] 12 8 [21, 3, 18] 13 10 [24, 3, 21] 14
11 [27, 3, 24] 15 10 [21, 3, 21] 16 7 [18, 3, 15] 17 13 [30, 3, 27] 18 11 [27, 3, 24] 19 10 [18, 3, 21] 20 11 [27, 3, 24] 21 13 [30, 3, 27] 22
The cubic curve in
Table 8, the points of PG(2, 23)
on :
Table 9.
The points of PG(2, 23) on
The generator matrix M of
= is:
Such that for all x
in with
, represents rows of M and
Let, = . Since D ([1, 0, 0]) = D ([0, 1, 0]) = D ([1, 0]) = 0, so [1, 0, 0], [0, 1, 0] and [ , 1, 0] are inflexion points of
IX.
S
UMMARY OFT
HISR
ESEARCHIn Table 10, a cell means that n is the number of points on the curve and is the number of distinct curves. The results we have obtained in this research are shown in Table 10 below.
Table 10.
Numbers of projectively distince cubics
Zero inflexions One inflexion
Three inflexions q
68 22 24 22
A
CKNOWLEDGMENTSI would like to thank and appreciate my supervisor Dr. Najm Abdulzahra, who directed this research, who did not reserve any effort to support me and his valuable advices throughout the research period, I am thankful to all my professors in the Department of Mathematics, College of Science, Mustansiriyah University for their positive support for researchers.
