IHJPAS. 36(2)2023 

375 
This work is licensed under a Creative Commons Attribution 4.0 International License. 

 

   

 

 

 

 

 
 

 

 

 
 

 

 

Abstract 

A cap of size 𝑘 and degree 𝑟 in a projective space, (briefly; (𝑘, 𝑟)-cap) is a set of 𝑘 points with 

the property that each line in the space meet it in at most 𝑟 points. The aim of this research is to 

extend the size and degree of complete caps and incomplete caps, (𝑘, 𝑟)-caps of degree 𝑟 < 12 in 

the finite projective space of dimension three over the finite field of order eleven, which already 

exist and founded by the action of subgroups of the general linear group over the finite field of 

order eleven and degree four, to (𝑘 + 𝑖, 𝑟 + 1)-complete caps. These caps have been classified by 

giving the 𝑡𝑖-distribution and 𝑐𝑖-distribution. The Gap programming has been used to execute the 

designed algorithms and computations.  

 

Keywords: Cap, Complete (Incomplete) cap, Companion matrix, Projective space, 𝜏𝑖- 

distribution, 𝑐𝑖-distribution. 

 

1.  Introduction 

 

In finite projective space, one of the important objects which has been many mathematicians did 

research on was (𝑘, 𝑟)-cap. The operator 𝑘 is the size of cap, and 𝑟 is the degree of cap.  These 

operators were the central of research, theoretically and by computer computations, where the 

researcher attempted to find the upper bound and lower bound of the size 𝑘 for fixed degree 𝑟. 

Also, the classification of the caps as projectively distinct was their aim, but this goal is so difficult 

in the space of dimension greater that two.  

 

Let 𝐹11 = {0,1,2, 2
2, … , 29} be the Galois field of order eleven, and 𝐶𝑓 be the 4 × 4 matrix, 

𝐶𝑓=(

0 1
0 0

0 0
1  0

0 0
28 2

 
0 1
1 22

) which is called the companion matrix over 𝐹11. The points of the finite  

doi.org/10.30526/36.2.3025 

Article history: Received 19 September 2022, Accepted 29 November 2022, Published in April 2023. 

 

 

 

Ibn Al-Haitham Journal for Pure and Applied Sciences 

Journal homepage: jih.uobaghdad.edu.iq 

 

Extension of Cap by Size and Degree in the Space 𝑷𝑮(𝟑, 𝟏𝟏) 
 

Jabbar Sharif Radhi 
Department of Mathematics, College of 

Science, Mustansiriyah University, 

Baghdad, Iraq. 

e.b.abdulkrareem@uomustansiriyah.edu.iq 

 

 

Emad Bakr Al-Zangana 

Department of Mathematics, College of 

Science, Mustansiriyah University, 

Baghdad, Iraq. 

jabbarsharif13@gmail.com 

https://creativecommons.org/licenses/by/4.0/
mailto:e.b.abdulkrareem@uomustansiriyah.edu.iq
mailto:jabbarsharif13@gmail.com


IHJPAS. 36(2)2023 

376 
 

projective space Σ11 = 𝑃𝐺(3,11) are constructing using 𝐶𝑓 in the formula 𝑃(𝑖) ≔ [1,0,0,0]𝐶𝑓
𝑖−1

, 

𝑖 = 1,2, … ,1464, where 1464 = 𝜃(3,11) =  113 + 112 + 11 + 1 is the order of space. This 

matrix is cyclic of order 𝜃(3,11), so to refer to the space’s points we can use numeral form 𝑖 ≔

𝑃(𝑖), 𝑖 = 1, … 𝜃(3,11). The lines in the space are found using the following formula: 

Let 𝑋 = [𝑥0, 𝑥1, 𝑥2, 𝑥3] and 𝑌 = [𝑦0, 𝑦1, 𝑦2, 𝑦3] be two point in Σ11 on a line 𝑙. A coordinate vector 

of 𝑙 is 𝐿 = (𝑙01, 𝑙02, 𝑙03, 𝑙12, 𝑙31, 𝑙23), where 𝑙𝑖𝑗 = 𝑥𝑖 𝑦𝑗 − 𝑥𝑗 𝑦𝑖.  Then 𝐿 is determined by 𝑙 up to a 

factor of proportion. We will write  𝑙 = 𝐈(𝐿) to refer to 𝑙. To find points 𝑍 = [𝑧0, 𝑧1, 𝑧2, 𝑧3] on 𝑙, 

the following equation must be satisfied [1]:  

𝑍 × (

0 −𝑙23
𝑙23 0

−𝑙31 −𝑙12
−𝑙03  𝑙02

𝑙31 𝑙03
𝑙12 −𝑙02

 
0 −𝑙01

𝑙01 0

) = 0. 

 

Definition 1.1:[2] A(𝑘, 𝑟)-cap in 𝑃𝐺(𝑛 ≥ 3, 𝑞) is the set of 𝑘 points such that no 𝑟 + 1   points are  

                        collinear, but at most 𝑟 points of which lie in any line. Here 𝑟 is called degree of   

                        the (𝑘, 𝑟)-cap. The (𝑘, 𝑟)-cap is called complete cap if it is not contained in (𝑘 + 

                           1, 𝑟)-cap. 

 

Definition 1.2:[2] Let 𝐾 be a cap of degree 𝑟, an 𝑖-secant of a 𝐾 in 𝑃𝐺(𝑛, 𝑞) is a line such that |𝑘 ∩

𝜋| = 𝑖. The number of 𝑖-secants of 𝐾 denoted by 𝜏𝑖. 

 

Definition 1.3:[2] Let 𝑄 be a point not on the (𝑘, 𝑟)-cap, 𝐾. The number of 𝑖-secant of 𝐾 passing 

through 𝑄 denoted by 𝜎𝑖 (𝑄). The number 𝜎𝑟 (𝑄) of 𝑟-secants is called the index of 𝑄 with respect 

to 𝐾.  

 

Definition 1.4:[2] The set of all points of index 𝑖 will be denoted by 𝐶𝑖 and the cardinality of 𝐶𝑖 

denoted by 𝑐𝑖. The sequence (𝑡0 , … , 𝑡𝑟 ) will represented the secant distribution and the sequences 

(𝑐0, … , 𝑐𝑑) refer to index distribution. 

 

Definition 1.5:[2] The group of projectivities of  𝑃𝐺(𝑛. 𝑞) is called projective general linear group, 

and denoted 𝑃𝐺𝐿 (𝑛 + 1, 𝑞). The elements of 𝑃𝐺𝐿(𝑛 + 1, 𝑞) are non-singular matrices of 

dimension 𝑛 + 1. The matrix  𝐶𝑓 is belong to Σ11, so the 𝐻 = 〈𝐶𝑓  〉 is cyclic subgroup of 

𝑃𝐺𝐿(4,11) of order 𝜃(3,11), and for any integer 𝑡 divided 𝜃(3,11),  𝐻𝑡 = 〈𝐶𝑓
𝑡
 〉 is cyclic subgroup 

of 𝐻 of order 𝑘 such that 𝑡 ∙ 𝑘 = 𝜃(3,11).  

 [3] found caps using the action of fourteen subgroups 𝐻𝑡 on the space Σ11, where 𝑡 =

2,3,4,6,8,12,24,61,122,183,244,366,488,732. These caps were just orbits come from action of 

the groups 𝐻𝑡 on Σ11, and these orbits denoted by 𝑂𝑡 [𝑡, 𝑘]. Also, they classified these caps by 

secant distribution and index distribution. Their results are summarized as in Table 1. Let 

INCO:=Incomplete cap, and CO:=Complete cap. 

 

 

 

 

 



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377 
 

Table 1. Caps from subgroups 〈𝑇𝑡 〉 action on 𝑃𝐺(3,11) 

 〈𝑇𝑡 〉 (𝑘, 𝑑𝑒𝑔𝑟𝑒𝑒 
𝑜𝑓 𝑐𝑎𝑝) 

𝑂𝑡 [𝑡, 𝑘] Type 

1 〈𝑇2〉 (732,12) {2𝑛 + 1|𝑛 = 0, … ,731} INCO 
2 〈𝑇3〉 (488,7) {3𝑛 + 1|𝑛 = 0, … ,487} CO 
3 〈𝑇4〉 (366,6) {4𝑛 + 1|𝑛 = 0, … ,365} CO 
4 〈𝑇6〉 (244,4) {6𝑛 + 1|𝑛 = 0, … ,243} CO 
5 〈𝑇8〉 (183,4) {8𝑛 + 1|𝑛 = 0, … ,182} INCO 
6 〈𝑇12〉 (122,2) {12𝑛 + 1|𝑛 = 0, … ,121} CO 
7 〈𝑇24〉 (61,2) {24𝑛 + 1|𝑛 = 0, … ,60} INCO 
8 〈𝑇61〉 (24,12) {61𝑛 + 1|𝑛 = 0, … ,23} INCO 
9 〈𝑇122〉 (12,12) {122𝑛 + 1|𝑛 = 0, … ,11} INCO 
10 〈𝑇183〉 (8,4) {183𝑛 + 1|𝑛 = 0, … ,7} INCO 
11 〈𝑇244〉 (6,6) {244𝑛 + 1|𝑛 = 0, … ,5} INCO 
12 〈𝑇366〉 (4,4) {366𝑛 + 1|𝑛 = 0, … ,3} INCO 
13 〈𝑇488〉 (3,3) {488𝑛 + 1|𝑛 = 0, … ,2} INCO 
14 〈𝑇732〉 (2,2) {732𝑛 + 1|𝑛 = 0,1} INCO 

 

The goal of this paper is to do extensions of the size of the orbits 𝑂𝑖 [𝑖, 𝑡] and compute the 𝜏𝑖-

distribution and 𝑐𝑖-distribution to it. 

Concerning the recent works on the caps resulting from the action of groups on the space 𝑃𝐺(3, 𝑞), 

there are some papers appear recently for example, [4,5], where they just find caps which exactly 

the orbits of group actions. The same idea has been used on the plane [6], and on the line 𝑃𝐺(1,27) 

[7]. 

    To do extension of the size of (complete, incomplete) (𝑘, 𝑟)-caps, 𝑂𝑡 [𝑡, 𝑘] in the projective  

space 𝑃𝐺(3,11), we follow the following steps:  

1. Determine the line 𝐈(𝐿) that meets the orbit 𝑂𝑖 [𝑖, 𝑡] in 𝑟 points. 

2. Find the extension points to 𝑂𝑖 [𝑖, 𝑡] which are the set 𝐿\𝑂𝑖 [𝑖, 𝑡] = 𝐸𝑥.  

3. Adding the first point say 𝑝1 of 𝐸𝑥 to  𝑂𝑖 [𝑖, 𝑡]. Let 𝑄𝑖
𝑝1 = 𝑂𝑖 [𝑖, 𝑡]⋃{𝑝1}. 

4. Check the set 𝑄𝑖
𝑝1  is (complete, incomplete) cap. 

5. Find the 𝜏𝑖-distribution and 𝑐𝑖-distribution of 𝑄𝑖
𝑝1 . 

6. The incomplete caps 𝑄𝑖
𝑝1  are extended to complete caps. 

A system for computational discrete algebra, GAP [8] is used to implement the above steps. 

Let 𝑌 = {2,3,4,6,8,12,24,61,122,183,244,366,488,732} be the divisors of 𝜃(3,11). 

2. Extension of Caps 

Theorem 2.1: Let 𝑂𝑖 [𝑖, 𝑡] be the (𝑖, 𝑟)-caps as in Table 1,  𝑖, 𝑡 in 𝑌 and 𝑟 ≠ 12.  Then there exist 

(𝑗, 𝑟 + 1)-complete caps, 𝑟 = 2,3,4,6,7 and = 636,505,313,304,139,90,260,650, 267,218,118.  

Proof: To do an extension of the size of each (𝑖, 𝑟)-cap we first find a line that meets the cap in 𝑟- 

            points, and then choose the first point belong to  𝐿\𝑂𝑖 [𝑖, 𝑡]. If the new cap is in complete,  

            then we will add points of the index zero points to make it complete. 

 

1. The line 𝐈(1,1,0,0,0,0) meet the cap 𝑂3[3,488] in 7 points, so we have five extension points can 

be adding to it in the following orders: 95,287,308,1167,1172. Let 𝑄3
𝑝1 = 𝑂3[3,488]⋃{𝑝1}, 𝑝1 =

95. The 𝑡𝑖 -distribution is (𝑡1 , 𝑡 2,  𝑡3 ,  𝑡4 ,  𝑡5 ,  𝑡6 , 𝑡 7 , 𝑡8) =(965,2909,997,6422,1022, 

2917,989,5 ), so  𝑂3
𝑝1  is (489,8)-cap and 𝑐𝑖 values are  𝑐0 = 955, 𝑐1 = 20. Since 𝑐0 ≠ 0, 



IHJPAS. 36(2)2023 

378 
 

then 𝑂3
𝑝1   is incomplete. The (489,8)-cap will be complete when adding 147 points to it, and the 

points are 2,3, 5, 6, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 26, 27, 29, 30, 32, 33, 35, 36, 38, 

39, 41, 42, 44, 45, 47, 48, 50, 51, 53, 54, 56, 57, 59, 60, 62, 63, 65, 66, 68, 69, 71, 72, 74, 75, 77, 

78, 80, 81, 83, 84, 86, 87, 89, 90, 92, 93, 95, 96, 98, 99, 101, 102, 104, 105, 107, 108, 110, 111, 

113, 114, 116, 117, 119, 120, 122, 123, 125, 126,128, 129, 131, 132, 134, 135, 137, 138, 140, 141, 

143, 144, 146, 147, 149, 150, 152, 153, 155, 156, 158, 159, 161,162, 164, 165, 167, 168, 170, 171, 

173, 174, 176, 177, 179, 180, 182, 183, 185, 186, 188, 189, 191, 192, 194, 195,197, 198, 200, 201, 

203, 204, 206, 207, 209, 210, 212, 213, 215, 216, 218, 219, 221, 222. 

 

2. The line 𝐈(25, 1,0,0,0,0) meet the cap 𝑂4[4,366] in 6 points, so we have six extension points 

can be  adding to it in the following orders: 239,346,542,766,955,1058. Let 𝑂4
𝑝1 = 

𝑂4[4,366]⋃{𝑝1},  𝑝1=239. The 𝑡𝑖-distribution is ( 𝜏0, 𝜏1 , 𝜏 2,  𝜏3 ,  𝜏4 ,   𝜏5 ,  𝜏6 , 𝜏 7 )= 

(785,1460,5082,1506,5106 1482,800,5), so 𝑂4
𝑝1  is (367,7)-cap and 𝑐𝑖 values are 𝑐0 = 1077,

𝑐1 = 20. Since 𝑐0 ≠ 0, then 𝑂4
𝑝1   is incomplete. The (367,7)-cap will be complete when adding 

138 points to it and the points are 2, 3, 4, 6, 7, 8, 10, 11, 12, 14, 15, 16, 18, 19, 20, 22, 23, 24, 26, 

27, 28, 30, 31, 32, 34, 35, 36, 38, 39, 40, 42, 43, 44, 46, 47, 48, 50, 51, 52, 54, 55, 56, 58, 59, 60, 

62, 63, 64, 66, 67, 68, 70, 71, 72, 74, 75, 76, 78, 79, 80, 82, 83, 84, 86, 87, 88, 90, 91, 92, 94, 95, 

96, 98, 99, 100, 102, 103, 104, 106, 107, 108, 110, 111, 112, 114,115, 116, 118, 119, 120, 122, 

123, 124, 126, 127, 128, 130, 131, 132, 134, 135, 136, 138, 139, 140, 142, 143, 144,146, 147, 148, 

150, 151, 152, 154, 155, 156, 158, 159, 160, 162, 163, 164, 166, 167, 168, 170, 171,172, 174, 

175,176, 178, 179, 180, 182, 183, 184.  

 

3. The line 𝐈(28, 26, 1,0,0,0) meet the cap 𝑂6[6,244] in 4 points, so we have eight extension points 

can be adding to it in the following orders: 16,338,422,716,776,891,988,455. Let 𝑂6
𝑝1 = 

𝑂6[6,244]⋃{𝑝1},  𝑝1 = 16. The 𝑡𝑖-distribution is ( 𝑡0, 𝑡1, 𝑡 2,  𝑡3 ,  𝑡4 , 𝑡5 )= 

(2958,1485,7270,1510,2985,18), so  𝑂6
𝑝1     is (245,5)-cap and 𝑐𝑖 values are 𝑐0 = 1093, 𝑐1 =

126. Since 𝑐0 ≠ 0, then 𝑂6
𝑝1  is incomplete. The (245,5)-cap will be complete  when adding 68 

points to it, and the points are 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 26, 

27, 28, 29, 30, 32, 33, 34, 35, 36, 38,39, 40, 41, 42, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 56, 57, 

58, 59, 60, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 80, 81, 82, 83.  

 

4. The line 𝐈(27, 26, 1,0,0,0) meet the cap 𝑂8[8,183] in 4 points, so we have eight extension points 

can be adding to it in the following orders: 318,860,1032,1158,1176,1252,1287,1444. Let 𝑂8
𝑝1 = 

𝑂8[8,183]⋃{𝑝1}, 𝑝1=318. The 𝑡𝑖-distribution is ( 𝑡0, 𝑡1 , 𝑡2,  𝑡3 , 𝑡4, 𝑡5)= 

(3683,4394,5487,1553,1100,9), so 𝑂8
𝑝1  is (184,5)-cap and 𝑐𝑖 values are 𝑐0 = 1217, 𝑐1 = 63. 

Since 𝑐0 ≠ 0, then 𝑂8
𝑝1  is incomplete. The (184,5)-cap will be complete when adding 120 points 

to it, and the points are 2,3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 26, 

27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 

56, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 

86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 101, 102, 103, 104, 106, 107, 108, 109, 110, 

111, 112, 114, 115, 116, 117, 118, 119, 120, 122, 123, 124, 125, 126, 127,128, 130, 131, 132, 133, 

134, 135, 136, 138.  

 



IHJPAS. 36(2)2023 

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5. The line 𝐈(1,0,1,0,0,0) meet the cap 𝑂12[12,122] in 2 points, so we have ten extension points 

can be  adding to it in the following orders:2,58,120,144,315,1252,452,506,788, 1108,1426. Let 

𝑂12
𝑝1 =𝑂12[12,122]⋃{𝑝1}, 𝑝1 = 2. The 𝑡𝑖-distribution is (𝑡0, 𝑡1, 𝑡2, 𝑡3) =(7315,1518,7338,55), 

so 𝑂12
𝑝1  is (123,3)-cap and 𝑐𝑖 values are 𝑐0 = 846, 𝑐1 = 495. Since 𝑐0 ≠ 0, then 𝑂12

𝑝1  is 

incomplete. The (123,3)- cap will be complete when adding 16 points to it, and the points are 3, 

4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19.     

 

6. The line 𝐈(1,0,1,0,0,0) meet the cap 𝑂24[24,61] in 2 points, so we have ten extension points can 

be adding to it in the following orders: 2,58,120,144,315,1252,452,506,788,1108, 1426. Let 

𝑂24
𝑝1 =𝑂24[24,161]⋃{𝑝1}, 𝑝1 = 2. The 𝑡𝑖-distribution is (𝑡0, 𝑡1, 𝑡2, 𝑡3) = (9859,4500,1855,12), 

so 𝑂24
𝑝1  is (25,3)-cap and 𝑐𝑖 values are 𝑐0 = 1294, 𝑐1 = 108. Since 𝑐0 ≠ 0, then 𝑂24

𝑝1  is 

incomplete. The (25,3)-cap will be complete when adding 65 points to it, and the points are 3, 4, 

5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 

34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 

61, 62, 63 64, 65, 66,67,68,69. 

 

7. The line 𝐈(22, 23, 1,0,0,0) meet the cap 𝑂183[183,8] in 4 points, so we have eight extension 

points can be  adding to it in the following orders: 123,245,611,855,315,977,1221,1343. Let 

𝑂183
𝑝1 =𝑂183[183,8]⋃{𝑝1}, 𝑝1 = 123. The 𝑡𝑖-distribution of this orbit is (𝑡0 , 𝑡1, 𝑡2, 𝑡4, 𝑡5) 

= (15056, 1148, 20,1,1), so 𝑂183
𝑝1  is (9,5)-cap and 𝑐𝑖 values are 𝑐0 = 1448, 𝑐1 = 7. Since 𝑐0 ≠

0, then 𝑂183
𝑝1  is incomplete. The (9,5)-cap will be complete when adding  251 points to it, and the 

points are 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 

28, 29, 30, 31, 32,33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 

54, 55, 56, 57, 58, 59, 60, 61,62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 

80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 

104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 

124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 

143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 

162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 

181, 182, 183, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 

201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 

220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 

239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254. 

 

8. The line 𝐈(22, 23, 1,0,0,0) meet the cap 𝑂244[244,6] in 6 points, so we have six extension points 

can be  adding to it in the following orders:123,367,611,855,1099,1343. Let 𝑄244
𝑝1 = 

𝑂244[244,6]⋃{𝑝1}, 𝑝1=123. The 𝑡𝑖 -distribution of this orbit is (𝑡0 , 𝑡1, 𝑡7) = (15301, 924, 1), 

so 𝑂183
𝑝1  is (7,7)-cap and 𝑐𝑖 values are 𝑐0 = 1452, 𝑐1 = 5. Since 𝑐0 ≠ 0, then 𝑂244   

𝑝1 is incomplete 

The (7,7)-cap will be complete when adding  515 points to it And the points are 2, 3, 4, 5, 6, 7, 8, 

9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31,32, 33, 34, 35, 

36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59,60, 61, 

62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87 

88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 

110, 111, 112 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 124, 125, 126, 127, 128, 129, 130, 



IHJPAS. 36(2)2023 

380 
 

131, 132, 133, 134, 135, 136,137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 

151, 152, 153, 154, 155, 156, 157, 158, 159,160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 

171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182,183, 184, 185, 186, 187, 188, 189, 190, 

191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 

210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 2224, 225, 226, 227, 228, 229, 

230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 246, 247, 248, 249, 

250, 251, 252,  253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 

269, 270, 271, 272, 273, 274, 275,  276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 

288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 

307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321,  322, 323, 324, 325, 

326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344,  

345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 

364, 365, 366, 367,  368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382, 

383, 384, 385, 386, 387, 388, 389, 390, 391, 392, 393, 394, 395, 396, 397, 398, 399, 400, 401, 

402, 403, 404, 405, 406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 420, 

421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435, 436, 437, 438, 439, 

440, 441, 442, 443, 444, 445, 446, 447, 448, 449, 450, 451, 452, 453, 454, 455, 456, 457, 458, 

459,460, 461, 462, 463, 464, 465, 466, 467, 468, 469, 470, 471, 472, 473, 474, 475, 476, 477, 478, 

479, 480, 481, 482, 483, 484, 485, 486, 487, 488, 490, 491, 492, 493, 494, 495, 496, 497, 498, 

499, 500, 501, 502, 503, 504, 505, 506,507, 508, 509, 510, 511, 512, 513, 514, 515, 516, 517, 518. 

      

9. The line 𝐈(22, 23, 1,0,0,0) meet the cap 𝑂366[366,4] in 4 points, so we have eight extension 

points can be  adding to it in the following orders: 123,245,489,611,855,977,1221,1343. Let 𝑂366
𝑝1 = 

𝑂366[366,4]⋃{𝑝1}, 𝑝1=123. The  𝑡𝑖-distribution is (𝑡0 , 𝑡1, 𝑡5) =(15565, 660, 1), so 𝑂366
𝑝1  is (5,5)-

cap and 𝑐𝑖 values are 𝑐0 = 1452,  𝑐1 = 7.  Since  𝑐0 ≠ 0, then 𝑂366
𝑝1 is incomplete. The (5,5) will 

be complete when adding 262 points to it, and the points are 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 

14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32,33, 34, 35, 36, 37, 38, 39, 

40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61,62, 63, 64, 65, 

66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90,91, 

92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 

113, 114, 115,116, 117, 118, 119, 120, 121, 122, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 

134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 

153, 154, 155, 156, 157, 158, 159, 160, 161, 162,163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 

173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185,186, 187, 188, 189, 190, 191, 192, 

193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 

212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 

231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 

250, 251, 252, 253, 254,255, 256, 257, 258, 259, 260, 261, 262, 263, 264. 

 

10. The line 𝐈(22, 23, 1,0,0,0) meet the cap 𝑂488[488,3] in 3 points, so we have nine extension 

points can be  adding to it in the following orders 123,245,376,611, 733,855, 1099,1221,1343. Let 

𝑂488
𝑝1 = 𝑂488[488,3]⋃{𝑝1}, 𝑝1=123. The 𝑡𝑖-distribution of this orbit is (𝑡0 , 𝑡1, 𝑡4)= (15697, 528, 1), 

so 𝑂488
𝑝1  is (4,4)-cap and𝑐𝑖 values are 𝑐0 = 1452, 𝑐1 = 8.  Since 𝑐0 ≠ 0, then 𝑂488 

𝑝1 is incomplete.  

The(4,4)-cap will be complete when adding 214 points to it from the index points 𝑐0 , and the 



IHJPAS. 36(2)2023 

381 
 

points are 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 

28, 29, 30, 31, 32,33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 

54, 55, 56, 57, 58, 59, 60, 61,  62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 

80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 

104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 

124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 

143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 

162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 

181, 182, 183, 184, 185,186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 

201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216. 

 

11. The line 𝐈(22, 23, 1,0,0,0) meet the cap 𝑂732[732,2] in 2 points, so we have ten extension 

points can be adding to it in the following orders: 123,245,367,489,611,855,977,1099, 1221,1343. 

Let 𝑂732
𝑝1 =𝑂732[732,2]⋃{𝑝1},  𝑝1=123. The 𝑡𝑖-distribution of this orbit is (𝑡0 , 𝑡1, 𝑡3) 

=(15829, 396, 1), so 𝑂732
𝑝1  is (3,3)-cap and 𝑐𝑖 values are 𝑐0 = 1452, 𝑐1 = 9.   Since 𝑐0 ≠ 0, then 

𝑂732
𝑝1 is incomplete. The(3,3)-cap will be complete when adding 115 points to it, and the points are 

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 

31, 3233, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 

57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 

83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 

107, 108, 109, 110, 111, 112, 113, 114, 115,116.  

 

3. Conclusion  

All the extension caps 𝑂3
𝑝1 , 𝑂4

𝑝1 , 𝑂6
𝑝1 , 𝑂8

𝑝1  𝑂12
𝑝1  , 𝑂24

𝑝1 , 𝑂183
𝑝1 , 𝑂244

𝑝1 ,𝑂366  
𝑝1 , 𝑂488

𝑝1 , 𝑂732
𝑝1  are incomplete   

caps, and can be completed by adding some points of index zero. The details are summarized 

inTable 2.   

Let # denoted to the number of adding points to the incomplete cap to be complete. 

 

Table 2. Details about the extension complete caps. 

 𝑂𝑖
𝑝1  𝜏𝑖 -distribution 𝑐𝑖 - distribution # 

1 𝑂3
𝑝1  

(𝜏1, 𝜏2  , 𝜏3 , 𝜏4 , 𝜏5 , 𝜏6, 𝜏7, 𝜏8) 

 
(𝑐0 , 𝑐1) 

147 

(965,2909,997,6422,1022,2917,989,5) (955,20) 

2 𝑂4
𝑝1  

(𝜏0, 𝜏1, 𝜏2  , 𝜏3 , 𝜏4 , 𝜏5 , 𝜏6, 𝜏7) 

 
(𝑐0 , 𝑐1) 

138 
(785,1460,5028,1506,5106, 

1482,800,5) 
(1077,20) 

3 𝑂6
𝑝1  

(𝜏0, 𝜏1, 𝜏2  , 𝜏3 , 𝜏4 , 𝜏5) 

 
(𝑐0 , 𝑐1) 

68 
(2958,1485,7270,1510 

, 2985,18) 

(1093,126) 

 

4 𝑂8
𝑝1  

(𝜏0 , 𝜏1 , 𝜏2, 𝜏3 , 𝜏4 , 𝜏5) 

 
(𝑐0 , 𝑐1) 

120 
( 3683,4394,5489, 

1553,1100,9) 
(1217,63) 

5 𝑂12
𝑝1  

(𝜏0, 𝜏1, 𝜏2  , 𝜏3 ) 

 
(𝑐0 , 𝑐1) 16 



IHJPAS. 36(2)2023 

382 
 

(7315,1518,7338,55) (846,495) 

6 𝑂24
𝑝1  

 

(𝜏0, 𝜏1, 𝜏2  , 𝜏3) 
(𝑐0 , 𝑐1) 

65 
(9859,4500,1855,12) 

 

(1294,108) 

7 𝑂183
𝑝1  

(𝜏0 , 𝜏1 , 𝜏2, 𝜏4 , 𝜏5) 

 

(𝑐0 , 𝑐1) 

 
251 

(15056,1148,20,1,1) 

 

(1448,7) 

8 𝑂244
𝑝1  

(𝜏0, 𝜏1, 𝜏7) (𝑐0 , 𝑐1) 

 
515 

(15301,924,1) 

 

(1452,5) 

9 𝑂366
𝑝1  

(𝜏0, 𝜏1, 𝜏5) (𝑐0 , 𝑐1) 

 

262  

(15565,660,1) 

 

(1452,7) 

 

10 𝑂488
𝑝1  

(𝜏0, 𝜏1, 𝜏4) (𝑐0 , 𝑐1) 

 
214 

(15697,528,1) (1452,8) 

 

11 𝑂732.
𝑝1  

(𝜏0, 𝜏1, 𝜏3) (𝑐0 , 𝑐1) 

 
115 

(15829,396,1) 

 

(1452,9) 

 

Acknowledgment  

         The authors would like to thanks the University of Mustansiriyah, department of Mathematics in   

         the College of Science for their motivation and support.  

 

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