Classification of <inline-formula><tex-math id="M1">\begin{document}$ \mathbf{(3 \!\mod 5)} $\end{document}</tex-math></inline-formula> arcs in <inline-formula><tex-math id="M2">\begin{document}$ \mathbf{ \operatorname{PG}(3,5)} $\end{document}</tex-math></inline-formula>

Sascha Kurz; Ivan Landjev; Assia Rousseva

doi:10.3934/amc.2021066

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

Sascha Kurz ^1, , Ivan Landjev ^2,3, and Assia Rousseva ^4,

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.

Keywords: Projective geometries, optimal linear codes, quasi-divisible arcs, $ (t\mod q) $-arcs, Griesmer bound.

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

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.
[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.
[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.
[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.
[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.
[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.
[7]	R. Hill, An extension theorem for linear codes, Designs, Codes and Cryptography, 17 (1999), 151-157. doi: 10.1023/A:1008319024396.
[8]	R. Hill and P. Lizak, Extensions of linear codes, in Proceedings of 1995 IEEE International Symposium on Information Theory, IEEE, 1995,345.
[9]	S. Kurz, Lecture Notes: Advanced and Current Topics in Coding Theory, 2020.
[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.
[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.
[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.
[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.
[14]	I. Landjev, A. 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.
[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.
[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.
[17]	G. Solomon and J. J. Stiffler, Algebraically punctured cyclic codes, Information and Control, 8 (1965), 170-179.

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.
[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.
[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.
[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.
[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.
[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.
[7]	R. Hill, An extension theorem for linear codes, Designs, Codes and Cryptography, 17 (1999), 151-157. doi: 10.1023/A:1008319024396.
[8]	R. Hill and P. Lizak, Extensions of linear codes, in Proceedings of 1995 IEEE International Symposium on Information Theory, IEEE, 1995,345.
[9]	S. Kurz, Lecture Notes: Advanced and Current Topics in Coding Theory, 2020.
[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.
[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.
[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.
[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.
[14]	I. Landjev, A. 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.
[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.
[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.
[17]	G. Solomon and J. J. Stiffler, Algebraically punctured cyclic codes, Information and Control, 8 (1965), 170-179.

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}(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 $

$ 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

$A_1$	$A_2$	$A_3$	$B_1$	$B_2$	$B_3$	$B_4$	$B_5$	$B_6$	$B_7$	$B_8$	$C_1$	$C_2$	$C_3$	$C_4$	$C_5$	$D_1$	$\lambda_0$	$\lambda_1$	$\lambda_2$	$\lambda_3$	$\#$
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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American Institute of Mathematical Sciences

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

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