The non-existence of (104,22;3,5)-arcs

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August 2016, 10(3): 601-611. doi: 10.3934/amc.2016029

The non-existence of $(104,22;3,5)$-arcs

Ivan Landjev ^1, and Assia Rousseva ^2,

1.	Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev str. bl. 8, Sofia 1113
2.	Faculty of Mathematics and Informatics, Sofia University, 5, James Bourchier blvd., 1164 Sofia, Bulgaria

Received May 2015 Revised September 2015 Published August 2016

Abstract
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Using some recent results about multiple extendability of arcs and codes, we prove the nonexistence of $(104,22)$-arcs in $PG(3,5)$. This implies the non-existence of Griesmer $[104,4,82]_5$-codes and settles one of the four remaining open cases for the main problem of coding theory for $q=5,k=4,d=82$.

Keywords: the Griesmer bound., extendable arcs, Projective geometries, optimal linear codes, quasidivisible arcs.

Mathematics Subject Classification: Primary: 51C05, 51E21, 51E22; Secondary: 94B0.

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

References:

[1]	S. Ball, On intersection sets in Desarguesian affine spaces, European J. Combin., 21 (2000), 441-446. doi: 10.1006/eujc.2000.0350.
[2]	I. Boukliev, Optimal Linear Codes - Constructions and Bounds, Ph.D thesis, Sofia, 1996.
[3]	S. Dodunekov and J. Simonis, Codes and projective multisets, Electr. J. Combin., 5 (1998), #R37.
[4]	Y. Edel and I. Landjev, On multiple caps in finite projective spaces, Des. Codes Cryptogr., 56 (2010), 163-175. doi: 10.1007/s10623-010-9398-4.
[5]	J. H. Griesmer, A bound for error-correcting codes, IBM J. Res. Develop., 4 (1960), 532-542.
[6]	R. Hill, Optimal linear codes, in Cryptography and Coding (ed. C. Mitchell), Oxford Univ. Press, 1992, 75-104.
[7]	R. Hill and E. Kolev, A survey of recent results on optimal linear codes, in Combinatorial Designs and their Application (eds. F.C. Holroyd, K.A.S. Quinn, Ch. Rowley and B.S. Webb), Chapman & Hall CRC, 1999, 127-152.
[8]	I. Landjev, The geometry of $(n,3)$-arcs in the projective plane of order 5, in Proc. 6th Workshop ACCT, Sozopol, 1996, 170-175.
[9]	I. Landjev and A. Rousseva, On the Extendability of Griesmer Arcs, Ann. Sof. Univ. Fac. Math. Inf., 101 (2013), 183-192.
[10]	I. Landjev, A. Rousseva and L. Storme, On the extendability of quasidivisible Griesmer arcs, Des. Codes Cryptogr., 79 (2016), 535-547. doi: 10.1007/s10623-015-0114-2.
[11]	I. Landjev and L. Storme, Linear codes and Galois geometries, in Current Research Topics in Galois Geometries (eds. L. Storme and J. De Beule), NOVA Publishers, 2012, 187-214.
[12]	T. Maruta, A new extension theorem for linear codes, Finite Fields Appl., 10 (2004), 674-685. doi: 10.1016/j.ffa.2004.02.001.
[13]	T. Maruta, http://www.mi.s.oskafu-u.ac.jp/~maruta/griesmer.htm
[14]	A. Rousseva, On the structure of $(t$ mod $q)$-arcs in finite projective geometries, Annuaire de l' Univ. de Sofia, 102 (2015), to appear.
[15]	G. Solomon and J. J. Stiffler, Algebraically punctured cyclic codes, Inform. Control, 8 (1965) 170-179.

show all references

References:

[1]	S. Ball, On intersection sets in Desarguesian affine spaces, European J. Combin., 21 (2000), 441-446. doi: 10.1006/eujc.2000.0350.
[2]	I. Boukliev, Optimal Linear Codes - Constructions and Bounds, Ph.D thesis, Sofia, 1996.
[3]	S. Dodunekov and J. Simonis, Codes and projective multisets, Electr. J. Combin., 5 (1998), #R37.
[4]	Y. Edel and I. Landjev, On multiple caps in finite projective spaces, Des. Codes Cryptogr., 56 (2010), 163-175. doi: 10.1007/s10623-010-9398-4.
[5]	J. H. Griesmer, A bound for error-correcting codes, IBM J. Res. Develop., 4 (1960), 532-542.
[6]	R. Hill, Optimal linear codes, in Cryptography and Coding (ed. C. Mitchell), Oxford Univ. Press, 1992, 75-104.
[7]	R. Hill and E. Kolev, A survey of recent results on optimal linear codes, in Combinatorial Designs and their Application (eds. F.C. Holroyd, K.A.S. Quinn, Ch. Rowley and B.S. Webb), Chapman & Hall CRC, 1999, 127-152.
[8]	I. Landjev, The geometry of $(n,3)$-arcs in the projective plane of order 5, in Proc. 6th Workshop ACCT, Sozopol, 1996, 170-175.
[9]	I. Landjev and A. Rousseva, On the Extendability of Griesmer Arcs, Ann. Sof. Univ. Fac. Math. Inf., 101 (2013), 183-192.
[10]	I. Landjev, A. Rousseva and L. Storme, On the extendability of quasidivisible Griesmer arcs, Des. Codes Cryptogr., 79 (2016), 535-547. doi: 10.1007/s10623-015-0114-2.
[11]	I. Landjev and L. Storme, Linear codes and Galois geometries, in Current Research Topics in Galois Geometries (eds. L. Storme and J. De Beule), NOVA Publishers, 2012, 187-214.
[12]	T. Maruta, A new extension theorem for linear codes, Finite Fields Appl., 10 (2004), 674-685. doi: 10.1016/j.ffa.2004.02.001.
[13]	T. Maruta, http://www.mi.s.oskafu-u.ac.jp/~maruta/griesmer.htm
[14]	A. Rousseva, On the structure of $(t$ mod $q)$-arcs in finite projective geometries, Annuaire de l' Univ. de Sofia, 102 (2015), to appear.
[15]	G. Solomon and J. J. Stiffler, Algebraically punctured cyclic codes, Inform. Control, 8 (1965) 170-179.

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