doi: 10.3934/amc.2021066
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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

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.

Citation: Sascha Kurz, Ivan Landjev, Assia Rousseva. Classification of $ \mathbf{(3 \!\mod 5)} $ arcs in $ \mathbf{ \operatorname{PG}(3,5)} $. Advances in Mathematics of Communications, doi: 10.3934/amc.2021066
References:
[1]

L. Baumert and R. McEliece, A note on the Griesmer bound, IEEE Transactions on Information Theory, 19 (1973), 134-135.  doi: 10.1109/tit.1973.1054939.  Google Scholar

[2]

I. Bouyukliev, S. Bouyuklieva and S. Kurz, Computer classification of linear codes, IEEE Trans. Inform. Theory, 67 (2021), 7807–7814. arXiv preprint, arXiv: 2002.07826, (2020), 18 pp.  Google Scholar

[3]

A. E. Brouwer and M. van Eupen, The correspondence between projective codes and $2$-weight codes, Designs, Codes and Cryptography, 11 (1997), 261-266.  doi: 10.1023/A:1008294128110.  Google Scholar

[4]

R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bulletin of the London Mathematical Society, 18 (1986), 97-122.  doi: 10.1112/blms/18.2.97.  Google Scholar

[5]

S. Dodunekov and J. Simonis, Codes and projective multisets, The Electronic Journal of Combinatorics, 5 (1998), Paper 37, 23 pp. doi: 10.37236/1375.  Google Scholar

[6]

J. H. Griesmer, A bound for error-correcting codes, IBM Journal of Research and Development, 4 (1960), 532-542.  doi: 10.1147/rd.45.0532.  Google Scholar

[7]

R. Hill, An extension theorem for linear codes, Designs, Codes and Cryptography, 17 (1999), 151-157.  doi: 10.1023/A:1008319024396.  Google Scholar

[8]

R. Hill and P. Lizak, Extensions of linear codes, in Proceedings of 1995 IEEE International Symposium on Information Theory, IEEE, 1995,345. Google Scholar

[9]

S. Kurz, Lecture Notes: Advanced and Current Topics in Coding Theory, 2020. Google Scholar

[10]

I. N. Landjev and A. P. Rousseva, On the extendability of Griesmer arcs, Annual of Sofia University "St. Kliment Ohridski" – Faculty of Mathematics and Informatics, 101 (2013), 183–192.  Google Scholar

[11]

I. Landjev and A. Rousseva, The non-existence of $(104, 22; 3, 5)$-arcs, Advances in Mathematics of Communications, 10 (2016), 601-611.  doi: 10.3934/amc.2016029.  Google Scholar

[12]

I. Landjev and A. Rousseva, On the characterization of $(3 \mod 5)$ arcs, Electronic Notes in Discrete Mathematics, 57 (2017), 187-192.  doi: 10.1016/j.endm.2017.02.031.  Google Scholar

[13]

I. Landjev and A. Rousseva, Divisible arcs, divisible codes, and the extension problem for arcs and codes, Problems of Information Transmission, 55 (2019), 226-240.  doi: 10.1134/s0555292319030033.  Google Scholar

[14]

I. LandjevA. Rousseva and L. Storme, On the extendability of quasidivisible Griesmer arcs, Designs, Codes and Cryptography, 79 (2016), 535-547.  doi: 10.1007/s10623-015-0114-2.  Google Scholar

[15]

T. Maruta, A new extension theorem for linear codes, Finite Fields and Their Applications, 10 (2004), 674-685.  doi: 10.1016/j.ffa.2004.02.001.  Google Scholar

[16]

A. Rousseva, On the structure of $(t \mod q)$-arcs in finite projective geometries, Annuaire de l'Univ. de Sofia, 102 (2015), 16 pp. Google Scholar

[17]

G. Solomon and J. J. Stiffler, Algebraically punctured cyclic codes, Information and Control, 8 (1965), 170-179.   Google Scholar

show all references

References:
[1]

L. Baumert and R. McEliece, A note on the Griesmer bound, IEEE Transactions on Information Theory, 19 (1973), 134-135.  doi: 10.1109/tit.1973.1054939.  Google Scholar

[2]

I. Bouyukliev, S. Bouyuklieva and S. Kurz, Computer classification of linear codes, IEEE Trans. Inform. Theory, 67 (2021), 7807–7814. arXiv preprint, arXiv: 2002.07826, (2020), 18 pp.  Google Scholar

[3]

A. E. Brouwer and M. van Eupen, The correspondence between projective codes and $2$-weight codes, Designs, Codes and Cryptography, 11 (1997), 261-266.  doi: 10.1023/A:1008294128110.  Google Scholar

[4]

R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bulletin of the London Mathematical Society, 18 (1986), 97-122.  doi: 10.1112/blms/18.2.97.  Google Scholar

[5]

S. Dodunekov and J. Simonis, Codes and projective multisets, The Electronic Journal of Combinatorics, 5 (1998), Paper 37, 23 pp. doi: 10.37236/1375.  Google Scholar

[6]

J. H. Griesmer, A bound for error-correcting codes, IBM Journal of Research and Development, 4 (1960), 532-542.  doi: 10.1147/rd.45.0532.  Google Scholar

[7]

R. Hill, An extension theorem for linear codes, Designs, Codes and Cryptography, 17 (1999), 151-157.  doi: 10.1023/A:1008319024396.  Google Scholar

[8]

R. Hill and P. Lizak, Extensions of linear codes, in Proceedings of 1995 IEEE International Symposium on Information Theory, IEEE, 1995,345. Google Scholar

[9]

S. Kurz, Lecture Notes: Advanced and Current Topics in Coding Theory, 2020. Google Scholar

[10]

I. N. Landjev and A. P. Rousseva, On the extendability of Griesmer arcs, Annual of Sofia University "St. Kliment Ohridski" – Faculty of Mathematics and Informatics, 101 (2013), 183–192.  Google Scholar

[11]

I. Landjev and A. Rousseva, The non-existence of $(104, 22; 3, 5)$-arcs, Advances in Mathematics of Communications, 10 (2016), 601-611.  doi: 10.3934/amc.2016029.  Google Scholar

[12]

I. Landjev and A. Rousseva, On the characterization of $(3 \mod 5)$ arcs, Electronic Notes in Discrete Mathematics, 57 (2017), 187-192.  doi: 10.1016/j.endm.2017.02.031.  Google Scholar

[13]

I. Landjev and A. Rousseva, Divisible arcs, divisible codes, and the extension problem for arcs and codes, Problems of Information Transmission, 55 (2019), 226-240.  doi: 10.1134/s0555292319030033.  Google Scholar

[14]

I. LandjevA. Rousseva and L. Storme, On the extendability of quasidivisible Griesmer arcs, Designs, Codes and Cryptography, 79 (2016), 535-547.  doi: 10.1007/s10623-015-0114-2.  Google Scholar

[15]

T. Maruta, A new extension theorem for linear codes, Finite Fields and Their Applications, 10 (2004), 674-685.  doi: 10.1016/j.ffa.2004.02.001.  Google Scholar

[16]

A. Rousseva, On the structure of $(t \mod q)$-arcs in finite projective geometries, Annuaire de l'Univ. de Sofia, 102 (2015), 16 pp. Google Scholar

[17]

G. Solomon and J. J. Stiffler, Algebraically punctured cyclic codes, Information and Control, 8 (1965), 170-179.   Google Scholar

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