Classification of $ \mathbf{(3 \!\mod 5)} $ arcs in $ \mathbf{ \operatorname{PG}(3,5)} $

Sascha Kurz; Ivan Landjev; Assia Rousseva

doi:10.3934/amc.2021066

Article Contents

2023, Volume 17, Issue 1: 172-206. Doi: 10.3934/amc.2021066

This issue Previous Article The interplay of different metrics for the construction of constant dimension codes Next Article On some codes from rank 3 primitive actions of the simple Chevalley group $ G_2(q) $

Classification of $ \mathbf{(3 \!\mod 5)} $ arcs in $ \mathbf{ \operatorname{PG}(3,5)} $

1.
Mathematisches Institut, Universität Bayreuth, D-95440 Bayreuth, Germany
2.
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 8 Acad G. Bonchev str., 1113 Sofia, Bulgaria
3.
New Bulgarian University, 21 Montevideo str, 1618 Sofia, Bulgaria
4.
Sofia University, Faculty of Mathematics and Informatics, J. Bourchier Blvd., 1164 Sofia, Bulgaria

Received: August 2021

Early access: December 2021

Published: December 2022

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Abstract

The proof of the non-existence of Griesmer $ [104, 4, 82]_5 $-codes is just one of many examples where extendability results are used. In a series of papers Landjev and Rousseva have introduced the concept of $ (t\mod q) $-arcs as a general framework for extendability results for codes and arcs. Here we complete the known partial classification of $ (3 \mod 5) $-arcs in $ \operatorname{PG}(3,5) $ and uncover two missing, rather exceptional, examples disproving a conjecture of Landjev and Rousseva. As also the original non-existence proof of Griesmer $ [104, 4, 82]_5 $-codes is affected, we present an extended proof to fill this gap.

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Mathematics Subject Classification: Primary: 51E22; Secondary: 51E21, 94B05.

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Table 1. Number of isomorphism types of strong $ (3\mod 5) $-arcs in $ \operatorname{PG}(2,5) $ and their corresponding minihypers

$ \# \mathcal{K} $	$ m $	$ \# \mathcal{B} $	line mult.	weights	# isomorphism types
18	3	3	$ 0,1,2,3 $	$ 0,1,2,3 $	4
23	4	9	$ 1,2,3,4 $	$ 5,6,7,8 $	1
28	5	15	$ 2,3,4,5 $	$ 10,11,12,13 $	1
33	6	21	$ 3,4,5,6 $	$ 15,16,17,18 $	10
38	7	27	$ 4,5,6,7 $	$ 20,21,22,23 $	23
43	8	33	$ 5,6,7,8 $	$ 25,26,27,28 $	53
48	9	39	$ 6,7,8,9 $	$ 30,31,32,33 $	49
53	10	45	$ 7,8,9,10 $	$ 35,36,37,38 $	17
58	11	51	$ 8,9,10,11 $	$ 40,41,42,43 $	11
63	12	57	$ 9,10,11,12 $	$ 45,46,47,48 $	9
68	13	63	$ 10,11,12,13 $	$ 50,51,52,53 $	6
73	14	69	$ 11,12,13,14 $	$ 55,56,57,58 $	0
78	15	75	$ 12,13,14,15 $	$ 60,61,62,63 $	0
83	16	81	$ 13,14,15,16 $	$ 65,66,67,68 $	0
88	17	87	$ 14,15,16,17 $	$ 70,71,72,73 $	0
93	18	93	$ 15,16,17,18 $	$ 75,76,77,78 $	1

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Table 2. Different line types of strong $ (3\mod 5) $-arcs in $ \operatorname{PG}(2,5) $

$ \mathcal{K}(L) $	type of $ L $	name
3	$ (3,0,0,0,0,0) $	$ A_1 $
	$ (2,1,0,0,0,0) $	$ A_2 $
	$ (1,1,1,0,0,0) $	$ A_3 $
8	$ (3,3,2,0,0,0) $	$ B_1 $
	$ (3,3,1,1,0,0) $	$ B_2 $
	$ (3,2,2,1,0,0) $	$ B_3 $
	$ (3,2,1,1,1,0) $	$ B_4 $
	$ (3,1,1,1,1,1) $	$ B_5 $
	$ (2,2,2,2,0,0) $	$ B_6 $
	$ (2,2,2,1,1,0) $	$ B_7 $
	$ (2,2,1,1,1,1) $	$ B_8 $
13	$ (3,3,3,3,1,0) $	$ C_1 $
	$ (3,3,3,2,2,0) $	$ C_2 $
	$ (3,3,3,2,1,1) $	$ C_3 $
	$ (3,3,2,2,2,1) $	$ C_4 $
	$ (3,2,2,2,2,2) $	$ C_5 $
18	$ (3,3,3,3,3,3) $	$ D_1 $

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Table 3. Details for the computations for $ 178\le \# \mathcal{K}\le 273 $

$ n $	178	183	188	193	198	203	208	213	218	223
$ \#\mathcal{E}_n $	31	36	46	75	180	174	176	179	177	179
time in h	3078	351	998	972	1434	1787	2368	2661	3214	3110
$ n $	228	233	238	243	248	253	258	263	268	273
$ \#\mathcal{E}_n $	176	180	177	170	176	170	161	173	148	111
time in h	3477	3448	3396	3150	2848	2042	1752	855	911	683

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Table 4. Strong $(3\mod 5)$-arcs in ${\rm{PG}}(2, 5)$ of cardinality $18$

$A_1$	$A_2$	$A_3$	$B_1$	$B_2$	$B_3$	$B_4$	$B_5$	$B_6$	$B_7$	$B_8$	$C_1$	$C_2$	$C_3$	$C_4$	$C_5$	$D_1$	$\lambda_0$	$\lambda_1$	$\lambda_2$	$\lambda_3$	$\#$
0	12	16	0	0	0	0	0	0	0	3	0	0	0	0	0	0	16	12	3	0	1
3	0	25	0	0	0	0	3	0	0	0	0	0	0	0	0	0	15	15	0	1	1
4	25	0	0	0	0	0	1	0	0	0	0	0	0	0	1	0	20	5	5	1	1
30	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	25	0	0	6	1

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Table 5. Strong $(3\mod 5)$-arcs in ${\rm{PG}}(2, 5)$ of cardinality $23$

$A_1$	$A_2$	$A_3$	$B_1$	$B_2$	$B_3$	$B_4$	$B_5$	$B_6$	$B_7$	$B_8$	$C_1$	$C_2$	$C_3$	$C_4$	$C_5$	$D_1$		$\lambda_0$	$\lambda_1$	$\lambda_2$	$\lambda_3$		$\#$
6	12	4	0	3	6	0	0	0	0	0	0	0	0	0	0	0		18	6	4	3		1

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Table 6. Strong $(3\mod 5)$-arcs in ${\rm{PG}}(2, 5)$ of cardinality $28$

$A_1$	$A_2$	$A_3$	$B_1$	$B_2$	$B_3$	$B_4$	$B_5$	$B_6$	$B_7$	$B_8$	$C_1$	$C_2$	$C_3$	$C_4$	$C_5$	$D_1$		$\lambda_0$	$\lambda_1$	$\lambda_2$	$\lambda_3$		$\#$
6	0	10	0	15	0	0	0	0	0	0	0	0	0	0	0	0		15	10	0	6		1

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Table 7. Strong $(3\mod 5)$-arcs in ${\rm{PG}}(2, 5)$ of cardinality $33$

$A_1$	$A_2$	$A_3$	$B_1$	$B_2$	$B_3$	$B_4$	$B_5$	$B_6$	$B_7$	$B_8$	$C_1$	$C_2$	$C_3$	$C_4$	$C_5$	$D_1$	$\lambda_0$	$\lambda_1$	$\lambda_2$	$\lambda_3$	$\#$
0	0	10	0	15	0	0	6	0	0	0	0	0	0	0	0	0	10	15	0	6	1
0	3	7	2	8	2	7	1	0	0	1	0	0	0	0	0	0	11	12	3	5	1
0	6	4	0	6	12	0	0	0	0	3	0	0	0	0	0	0	12	9	6	4	1
0	6	4	2	4	8	4	0	0	2	1	0	0	0	0	0	0	12	9	6	4	2
0	6	4	3	3	6	6	0	0	3	0	0	0	0	0	0	0	12	9	6	4	1
0	9	1	3	0	9	3	0	3	3	0	0	0	0	0	0	0	13	6	9	3	1
2	8	1	8	6	4	0	0	0	1	0	0	0	0	1	0	0	15	5	5	6	1
4	5	2	5	4	10	0	0	0	0	0	1	0	0	0	0	0	15	5	5	6	1
8	4	0	16	0	0	0	0	1	0	0	2	0	0	0	0	0	18	1	4	8	1

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Table 8. Strong $(3\mod 5)$-arcs in ${\rm{PG}}(2, 5)$ of cardinality $38$

$A_1$	$A_2$	$A_3$	$B_1$	$B_2$	$B_3$	$B_4$	$B_5$	$B_6$	$B_7$	$B_8$	$C_1$	$C_2$	$C_3$	$C_4$	$C_5$	$D_1$	$\lambda_0$	$\lambda_1$	$\lambda_2$	$\lambda_3$	$\#$
0	0	4	0	0	0	0	0	3	18	6	0	0	0	0	0	0	6	12	13	0	1
0	0	5	0	0	6	12	2	0	2	3	1	0	0	0	0	0	7	14	6	4	1
0	1	4	0	0	10	4	1	0	8	2	0	0	1	0	0	0	8	11	9	3	1
0	1	4	0	0	9	6	0	1	6	3	0	0	1	0	0	0	8	11	9	3	1
0	2	3	0	0	6	9	0	1	7	2	0	1	0	0	0	0	8	11	9	3	1
0	2	4	0	12	0	8	3	0	0	0	1	1	0	0	0	0	9	13	2	7	1
0	2	4	4	5	4	8	0	0	2	0	0	0	2	0	0	0	10	10	5	6	2
0	3	2	0	0	8	2	0	4	10	1	0	0	0	1	0	0	9	8	12	2	1
0	3	3	2	6	6	8	0	0	0	1	1	0	0	1	0	0	10	10	5	6	1
0	4	2	4	2	10	3	0	1	3	0	0	0	1	1	0	0	11	7	8	5	1
0	5	0	0	0	5	0	0	10	10	0	0	0	0	0	1	0	10	5	15	1	1
0	5	1	2	4	12	0	1	0	4	0	0	1	0	1	0	0	11	7	8	5	1
0	5	1	3	3	9	4	0	1	3	0	0	1	0	1	0	0	11	7	8	5	1
0	6	0	4	0	12	0	0	6	0	1	0	0	0	2	0	0	12	4	11	4	1
1	1	4	2	4	7	9	0	0	1	0	1	0	1	0	0	0	10	10	5	6	1
1	2	3	3	1	13	2	0	1	3	0	0	0	2	0	0	0	11	7	8	5	1
1	3	2	2	1	13	4	0	1	2	0	1	0	0	1	0	0	11	7	8	5	1
1	4	1	0	4	14	0	1	0	4	0	0	2	0	0	0	0	11	7	8	5	1
1	4	1	1	3	11	4	0	1	3	0	0	2	0	0	0	0	11	7	8	5	1
2	5	0	10	2	7	0	0	2	0	0	0	2	1	0	0	0	14	3	7	7	2
3	0	4	3	15	0	3	0	0	0	0	3	0	0	0	0	0	12	9	1	9	1

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Table 9. Strong $(3\mod 5)$-arcs in ${\rm{PG}}(2, 5)$ of cardinality $43$

$A_1$	$A_2$	$A_3$	$B_1$	$B_2$	$B_3$	$B_4$	$B_5$	$B_6$	$B_7$	$B_8$	$C_1$	$C_2$	$C_3$	$C_4$	$C_5$	$D_1$	$\lambda_0$	$\lambda_1$	$\lambda_2$	$\lambda_3$	$\#$
0	0	0	0	0	0	0	30	0	0	0	0	0	0	0	0	1	0	25	0	6	1
0	0	0	0	0	0	0	4	0	0	25	0	0	0	0	2	0	0	20	10	1	1
0	0	1	0	0	0	9	3	0	6	9	0	0	0	3	0	0	3	16	9	3	1
0	0	2	0	2	7	8	1	0	4	3	0	0	2	2	0	0	6	12	8	5	2
0	0	2	0	3	1	13	4	0	2	2	0	1	3	0	0	0	5	15	5	6	1
0	0	3	2	8	5	6	1	0	0	1	1	0	4	0	0	0	8	11	4	8	1
0	0	3	4	6	0	12	0	1	0	0	1	0	4	0	0	0	8	11	4	8	1
0	1	0	0	0	0	8	0	0	12	7	0	0	0	1	2	0	4	13	12	2	1
0	1	1	0	2	3	13	0	1	3	3	0	1	1	2	0	0	6	12	8	5	1
0	1	1	0	2	4	11	1	0	5	2	0	1	1	2	0	0	6	12	8	5	2
0	1	1	0	2	8	4	0	2	7	2	0	0	0	4	0	0	7	9	11	4	2
0	1	2	1	9	4	7	1	0	0	1	1	1	3	0	0	0	8	11	4	8	1
0	1	2	6	0	12	0	0	3	2	0	0	0	2	3	0	0	10	5	10	6	1
0	2	0	0	0	12	0	0	4	8	1	0	0	1	0	3	0	8	6	14	3	1
0	2	0	0	1	7	7	0	1	8	1	0	1	0	2	1	0	7	9	11	4	1
0	2	0	0	1	8	5	1	0	10	0	0	1	0	2	1	0	7	9	11	4	1
0	2	1	0	10	3	8	1	0	0	1	1	2	2	0	0	0	8	11	4	8	1
0	2	1	1	8	2	11	1	0	0	0	2	1	1	1	0	0	8	11	4	8	1
0	2	1	2	4	11	5	0	0	0	1	2	0	0	3	0	0	9	8	7	7	1
0	2	1	2	5	10	4	0	0	1	1	1	1	1	2	0	0	9	8	7	7	1
0	2	1	2	6	9	3	0	0	2	1	0	2	2	1	0	0	9	8	7	7	2
0	2	1	3	4	8	6	0	0	2	0	1	1	1	2	0	0	9	8	7	7	1
0	2	1	3	5	7	5	0	0	3	0	0	2	2	1	0	0	9	8	7	7	1
0	3	0	0	7	11	3	0	0	0	2	1	2	0	2	0	0	9	8	7	7	1
0	3	0	2	6	6	6	0	0	3	0	0	3	1	1	0	0	9	8	7	7	1
0	3	0	4	1	12	2	0	2	2	0	0	2	1	1	1	0	10	5	10	6	1
0	3	0	4	2	10	2	0	3	2	0	0	2	0	3	0	0	10	5	10	6	1
0	3	1	12	3	6	0	0	0	0	0	3	0	0	3	0	0	12	4	6	9	1
1	0	0	0	0	0	0	2	0	25	0	0	0	0	0	3	0	5	10	15	1	1
1	0	0	0	0	0	25	3	0	0	0	0	0	0	0	1	1	5	15	5	6	1
1	0	1	0	0	6	8	0	2	9	0	0	1	0	3	0	0	7	9	11	4	1
1	0	2	0	7	5	10	1	0	0	0	2	1	2	0	0	0	8	11	4	8	1
1	0	2	2	4	10	4	0	0	3	0	0	2	3	0	0	0	9	8	7	7	2
1	1	1	1	4	10	6	0	0	2	0	1	2	1	1	0	0	9	8	7	7	2
1	2	0	2	1	14	2	0	2	2	0	0	3	1	0	1	0	10	5	10	6	2
1	2	0	2	2	12	2	0	3	2	0	0	3	0	2	0	0	10	5	10	6	1
1	3	0	9	5	6	0	0	0	1	0	2	3	0	1	0	0	12	4	6	9	1
2	0	0	0	0	0	0	0	25	0	0	0	0	0	0	4	0	10	0	20	1	1
2	0	0	0	0	25	0	1	0	0	0	0	0	0	0	2	1	10	5	10	6	2
2	0	0	0	25	0	0	2	0	0	0	0	0	0	0	0	2	10	10	0	11	2
2	1	1	8	3	10	0	0	0	0	0	3	2	0	1	0	0	12	4	6	9	1
2	2	0	7	5	8	0	0	0	1	0	2	4	0	0	0	0	12	4	6	9	1
3	0	0	25	0	0	0	0	0	0	0	0	0	0	0	1	2	15	0	5	11	1
3	0	0	6	6	12	0	0	0	0	0	0	3	0	0	0	1	12	4	6	9	1

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Table 10. Strong $(3\mod 5)$-arcs in ${\rm{PG}}(2, 5)$ of cardinality $48$

$A_1$	$A_2$	$A_3$	$B_1$	$B_2$	$B_3$	$B_4$	$B_5$	$B_6$	$B_7$	$B_8$	$C_1$	$C_2$	$C_3$	$C_4$	$C_5$	$D_1$	$\lambda_0$	$\lambda_1$	$\lambda_2$	$\lambda_3$	$\#$
0	0	0	0	12	3	6	3	0	0	0	0	0	6	0	0	1	6	12	3	10	1
0	0	0	0	2	11	0	0	2	6	2	0	0	1	5	2	0	6	7	13	5	1
0	0	0	0	2	4	12	1	0	0	4	0	0	6	1	1	0	4	13	7	7	1
0	0	0	0	2	8	6	0	0	4	3	0	0	3	4	1	0	5	10	10	6	1
0	0	0	0	3	3	10	2	0	2	3	0	0	5	3	0	0	4	13	7	7	1
0	0	0	0	3	6	6	0	1	4	3	0	0	2	6	0	0	5	10	10	6	1
0	0	0	1	0	11	2	0	1	7	1	0	0	2	3	3	0	6	7	13	5	1
0	0	0	1	2	1	12	2	0	3	2	0	0	5	3	0	0	4	13	7	7	1
0	0	0	1	2	4	8	0	1	5	2	0	0	2	6	0	0	5	10	10	6	1
0	0	0	1	2	5	6	1	0	7	1	0	0	2	6	0	0	5	10	10	6	1
0	0	0	1	2	7	2	0	3	7	1	0	0	0	7	1	0	6	7	13	5	1
0	0	0	2	0	4	10	0	0	6	1	0	0	3	4	1	0	5	10	10	6	1
0	0	0	2	0	7	4	0	2	8	0	0	0	1	5	2	0	6	7	13	5	1
0	0	0	2	0	8	0	0	7	6	0	0	0	0	4	4	0	7	4	16	4	1
0	0	0	3	6	6	9	0	0	0	0	0	0	3	3	0	1	7	9	6	9	1
0	0	0	4	2	14	4	0	0	0	0	0	0	1	4	1	1	8	6	9	8	1
0	0	1	0	9	3	6	3	0	0	0	3	0	6	0	0	0	6	12	3	10	1
0	0	1	1	8	7	2	0	0	1	2	0	3	6	0	0	0	7	9	6	9	1
0	0	1	2	5	7	6	0	0	0	1	2	1	4	2	0	0	7	9	6	9	1
0	0	1	3	4	5	8	0	0	1	0	2	1	4	2	0	0	7	9	6	9	1
0	0	1	3	5	4	7	0	0	2	0	1	2	5	1	0	0	7	9	6	9	2
0	0	1	3	6	3	6	0	0	3	0	0	3	6	0	0	0	7	9	6	9	2
0	0	1	4	2	10	2	0	1	2	0	1	2	2	4	0	0	8	6	9	8	2
0	0	1	4	3	9	1	0	1	3	0	0	3	3	3	0	0	8	6	9	8	2
0	0	1	6	0	9	0	0	6	0	0	0	3	0	6	0	0	9	3	12	7	1
0	0	2	6	12	0	0	0	0	0	1	6	0	4	0	0	0	9	8	2	12	1
0	1	0	1	6	6	7	0	0	0	1	2	2	3	2	0	0	7	9	6	9	1
0	1	0	1	7	5	6	0	0	1	1	1	3	4	1	0	0	7	9	6	9	1
0	1	0	2	3	13	0	1	0	2	0	2	2	0	5	0	0	8	6	9	8	1
0	1	0	2	5	4	9	0	0	1	0	2	2	3	2	0	0	7	9	6	9	1
0	1	0	3	1	12	4	0	0	1	0	2	2	1	3	1	0	8	6	9	8	1
0	1	0	3	2	11	3	0	0	2	0	1	3	2	2	1	0	8	6	9	8	1
0	1	0	3	3	10	2	0	0	3	0	0	4	3	1	1	0	8	6	9	8	1
0	1	0	3	3	9	3	0	1	2	0	1	3	1	4	0	0	8	6	9	8	2
0	1	0	3	4	8	2	0	1	3	0	0	4	2	3	0	0	8	6	9	8	2
0	1	0	5	0	10	1	0	5	0	0	0	4	0	4	1	0	9	3	12	7	1
0	2	0	12	0	6	0	0	1	0	0	2	6	1	0	1	0	11	2	8	10	1
1	0	0	0	6	5	8	0	0	2	0	1	4	4	0	0	0	7	9	6	9	1
1	0	0	1	4	10	2	0	1	3	0	0	5	2	2	0	0	8	6	9	8	1
1	0	0	4	14	0	4	0	0	0	0	4	1	2	0	0	1	9	8	2	12	1
1	0	1	4	11	0	4	0	0	0	0	7	1	2	0	0	0	9	8	2	12	1
2	0	0	8	1	8	0	0	2	0	0	2	8	0	0	0	0	11	2	8	10	2

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Table 11. Strong $(3\mod 5)$-arcs in ${\rm{PG}}(2, 5)$ of cardinality $53$

$A_1$	$A_2$	$A_3$	$B_1$	$B_2$	$B_3$	$B_4$	$B_5$	$B_6$	$B_7$	$B_8$	$C_1$	$C_2$	$C_3$	$C_4$	$C_5$	$D_1$	$\lambda_0$	$\lambda_1$	$\lambda_2$	$\lambda_3$	$\#$
0	0	0	0	5	10	0	0	0	0	2	2	4	4	4	0	0	6	7	8	10	1
0	0	0	0	6	4	5	1	0	0	1	3	2	8	1	0	0	5	10	5	11	1
0	0	0	1	3	9	3	0	0	0	1	3	3	3	5	0	0	6	7	8	10	1
0	0	0	1	4	3	8	1	0	0	0	4	1	7	2	0	0	5	10	5	11	1
0	0	0	1	6	0	8	0	1	0	1	2	3	9	0	0	0	5	10	5	11	1
0	0	0	2	2	7	5	0	0	1	0	3	3	3	5	0	0	6	7	8	10	1
0	0	0	2	3	6	4	0	0	2	0	2	4	4	4	0	0	6	7	8	10	2
0	0	0	3	0	11	1	0	1	1	0	1	6	2	3	2	0	7	4	11	9	1
0	0	0	3	2	8	0	0	2	2	0	0	7	2	4	1	0	7	4	11	9	2
0	0	0	9	3	6	0	0	0	0	0	3	6	0	3	0	1	9	3	7	12	1
0	0	1	9	0	6	0	0	0	0	0	6	6	0	3	0	0	9	3	7	12	1
0	1	0	1	12	0	0	2	0	0	0	11	0	4	0	0	0	7	9	1	14	1
0	1	0	8	2	4	0	0	0	1	0	5	8	0	2	0	0	9	3	7	12	1
1	0	0	10	0	0	0	0	5	0	0	0	15	0	0	0	0	10	0	10	11	1
1	0	0	6	2	6	0	0	0	1	0	5	9	0	1	0	0	9	3	7	12	1

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Table 12. Strong $(3\mod 5)$-arcs in ${\rm{PG}}(2, 5)$ of cardinality $58$

$A_1$	$A_2$	$A_3$	$B_1$	$B_2$	$B_3$	$B_4$	$B_5$	$B_6$	$B_7$	$B_8$	$C_1$	$C_2$	$C_3$	$C_4$	$C_5$	$D_1$	$\lambda_0$	$\lambda_1$	$\lambda_2$	$\lambda_3$	$\#$
0	0	0	0	0	10	0	1	0	0	0	5	5	0	10	0	0	5	5	10	11	1
0	0	0	0	3	3	3	0	0	1	1	3	5	9	3	0	0	4	8	7	12	1
0	0	0	1	1	2	6	0	0	1	0	4	4	8	4	0	0	4	8	7	12	1
0	0	0	1	1	5	2	0	1	1	0	3	7	2	8	0	0	5	5	10	11	1
0	0	0	1	1	6	1	0	0	2	0	2	8	4	5	1	0	5	5	10	11	1
0	0	0	1	2	4	1	0	1	2	0	2	8	3	7	0	0	5	5	10	11	1
0	0	0	1	3	0	4	0	0	3	0	2	6	10	2	0	0	4	8	7	12	1
0	0	0	1	3	3	0	0	1	3	0	1	9	4	6	0	0	5	5	10	11	1
0	0	0	2	1	4	0	0	4	0	0	0	12	0	6	2	0	6	2	13	10	1
0	0	0	3	6	0	3	0	0	0	0	9	3	6	0	0	1	6	7	3	15	1
0	0	1	3	3	0	3	0	0	0	0	12	3	6	0	0	0	6	7	3	15	1

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Table 13. Strong $(3\mod 5)$-arcs in ${\rm{PG}}(2, 5)$ of cardinality $63$

$A_1$	$A_2$	$A_3$	$B_1$	$B_2$	$B_3$	$B_4$	$B_5$	$B_6$	$B_7$	$B_8$	$C_1$	$C_2$	$C_3$	$C_4$	$C_5$	$D_1$	$\lambda_0$	$\lambda_1$	$\lambda_2$	$\lambda_3$	$\#$
0	0	0	0	0	0	0	0	0	0	5	0	0	15	10	1	0	0	10	10	11	1
0	0	0	0	0	0	0	0	0	3	2	1	2	6	15	2	0	1	7	13	10	1
0	0	0	0	0	0	0	0	1	4	0	0	6	2	12	6	0	2	4	16	9	1
0	0	0	0	0	1	4	1	0	0	0	4	2	14	4	0	1	2	9	6	14	1
0	0	0	0	0	3	3	0	0	0	0	3	6	6	9	0	1	3	6	9	13	1
0	0	0	0	0	6	0	0	0	0	0	0	12	3	6	3	1	4	3	12	12	1
0	0	0	4	1	2	0	0	0	0	0	4	14	0	4	0	2	6	2	8	15	1
0	0	1	0	0	0	0	0	0	3	0	3	9	9	6	0	0	3	6	9	13	1
0	0	1	0	0	0	0	0	3	0	0	3	12	0	12	0	0	4	3	12	12	1

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Table 14. Strong $(3\mod 5)$-arcs in ${\rm{PG}}(2, 5)$ of cardinality $68$

$A_1$	$A_2$	$A_3$	$B_1$	$B_2$	$B_3$	$B_4$	$B_5$	$B_6$	$B_7$	$B_8$	$C_1$	$C_2$	$C_3$	$C_4$	$C_5$	$D_1$	$\lambda_0$	$\lambda_1$	$\lambda_2$	$\lambda_3$	$\#$
0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	30	1	0	0	25	6	1
0	0	0	0	0	0	0	1	0	0	0	0	0	0	25	3	2	0	5	15	11	1
0	0	0	0	0	0	0	2	0	0	0	0	0	25	0	1	3	0	10	5	16	1
0	0	0	0	3	0	0	0	0	0	0	6	6	12	0	0	4	3	6	4	18	1
1	0	0	0	0	0	0	0	0	0	0	0	25	0	0	2	3	5	0	10	16	1
1	0	0	0	0	0	0	1	0	0	0	25	0	0	0	0	4	5	5	0	21	1

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Table 15. Strong $(3\mod 5)$-arcs in ${\rm{PG}}(2, 5)$ of cardinality $93$

$A_1$	$A_2$	$A_3$	$B_1$	$B_2$	$B_3$	$B_4$	$B_5$	$B_6$	$B_7$	$B_8$	$C_1$	$C_2$	$C_3$	$C_4$	$C_5$	$D_1$		$\lambda_0$	$\lambda_1$	$\lambda_2$	$\lambda_3$		$\#$
0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	31		0	0	0	31		1

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