Some combinatorial aspects of pedal sets

Mustafa Gezek

doi:10.3934/amc.2025036

Article Contents

2026, Volume 20: 199-208. Doi: 10.3934/amc.2025036

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Some combinatorial aspects of pedal sets

Mustafa Gezek

Department of Mathematics, Tekirdağ Namık Kemal University, Tekirdağ, 59030, Türkiye

Received: January 2025

Revised: June 2025

Early access: August 8, 2025

Published online: August 8, 2025

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Abstract

Pedal sets serve as a useful tool for classifying unitals and can act as a foundation for discovering new unitals. In this article, it is proved that the numbers of pedal sets of line types $ (q+1) $ and $ (q, 2^q) $ in a unital $ U $ (embedded in a projective plane of order $ q^2 $) and its dual unital $ U^\perp $ are equal. Details of the statistics on the pedal sets of unitals in planes of order 25 are presented. Some counter-examples are given to disprove one of the previously proposed conjectures. A new unital embedded in one of the planes of order 25 is discovered. Distributions of collinear pedal sets not lying on a special tangent are analyzed. Computational results presented in this study not only address longstanding inquiries but also prompt new questions for future research directions.

Keywords:

Mathematics Subject Classification: 05B05, 51E10, 51E20.

Citation:

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Table 1. A new unital in the $ s4 $ plane

Plane	Unital No.	Unital
		8, 16, 22, 27, 31, 34, 45, 46, 59, 63, 67, 76, 79, 82, 99,
		104,106,108,109,113,122,123,134,136,137,138,
		142,146,148,159,163,166,172,179,181,193,197,
		198,199,209,214,216,219,225,228,233,235,239,
		240,245,249,260,268,269,271,278,280,292,294,
$ s4 $	11	323,324,326,327,331,335,337,338,339,341,342,
		348,350,352,384,395,396,398,402,407,408,415,
		419,424,425,434,436,440,441,443,450,452,453,
		459,460,463,467,476,482,493,499,502,507,516,
		525,529,532,534,540,557,562,567,574,578,592,
		598,604,606,607,615,617,624,636,638,641,648,
		650

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Table 2. The new unital design in the $ s4 $ plane

$ \|Aut(U)\| $	$ 2 $-rank	$ 3 $-rank	PC of $ U/U^\perp $	Par. PC of $ U/U^\perp $	Resolvable?
40	125	125	25/25	1050/1050	Yes

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Table 3. Counterexamples

Plane	$ a1b $	$ a2b $	$ a3a $	$ a6a $	$ b5b $	$ b7c $	$ s2b $	$ h1a $	$ w1 $
Unital No.	4	2	2,5	1	3	1	16,17	2	1,3

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Table 4. Some statistics related to outsider special points

Plane Order	Plane	Unitals	Number of SPs	Dist. of outsider SPs
9 [11]	HALL	2	15	$ 6(1) $
16 [7,11]	BBH1	1	32	$ 4(4) $
		2,14	68	-
		16	24	$ 4(2) $
	BBH2	19,20,22,23	24	$ 4(2) $
		21	32	$ 8(2) $
		26	40	$ 4(6) $
	HALL	4	32	$ 4(4) $
		6	68	$ 4(10) $,$ 12(1) $
	MATH	5	24	$ 4(2) $
		13	80	$ 4(16) $
	JOHN	2	32	$ 4(4) $
		26	24	$ 4(2) $
	SEMI4	1,4	20	$ 4(1) $
25	$ a1 $	5,6,9	30	$ 5(1) $
	$ a2 $	1	45	$ 20(1) $
	$ a2b $	2	35	$ 10(1) $
		10	30	$ 5(1) $
	$ a4 $	3	45	$ 20(1) $
		5	30	$ 5(1) $
	$ a5 $	3	35	$ 5(2) $
	$ a6 $	3	35	$ 5(2) $
		4	30	$ 5(1) $
	$ a7 $	7,8	30	$ 5(1) $
	$ a8 $	1,6	45	$ 20(1) $
	$ b3 $	1	30	$ 5(1) $
	$ b4 $	1	50	$ 5(3) $,$ 10(1) $
	$ b7 $	5,6	35	$ 10(1) $
	$ b8 $	1	45	$ 20(1) $
		2	45	$ 5(4) $
	$ s2 $	5	30	$ 5(1) $
		8	165	$ 5(12) $,$ 10(6) $,$ 20(1) $
	$ s2a $	1,2	45	$ 5(4) $
	$ s2b $	16	30	$ 5(1) $
	$ s3 $	2	105	$ 5(12) $,$ 20(1) $
	$ s4 $	3	45	$ 20(1) $
		7,9	30	$ 5(1) $
		10	70	$ 5(5) $,$ 10(2) $
	$ s5 $	2	50	$ 5(3) $,$ 10(1) $
		4,9	40	$ 5(3) $
		8	30	$ 5(1) $
		10	50	$ 5(1) $,$ 10(2) $
	$ h1a $	2	30	$ 5(1) $
	$ h2a $	1	33	$ 8(1) $
		2	35	$ 10(1) $
	$ w1 $	3	45	$ 20(1) $

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