# American Institute of Mathematical Sciences

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

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

 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

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 $q=5,k=4,d=82$.
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.  Google Scholar [2] I. Boukliev, Optimal Linear Codes - Constructions and Bounds, Ph.D thesis, Sofia, 1996. Google Scholar [3] S. Dodunekov and J. Simonis, Codes and projective multisets, Electr. J. Combin., 5 (1998), #R37.  Google Scholar [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.  Google Scholar [5] J. H. Griesmer, A bound for error-correcting codes, IBM J. Res. Develop., 4 (1960), 532-542.  Google Scholar [6] R. Hill, Optimal linear codes, in Cryptography and Coding (ed. C. Mitchell), Oxford Univ. Press, 1992, 75-104.  Google Scholar [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.  Google Scholar [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. Google Scholar [9] I. Landjev and A. Rousseva, On the Extendability of Griesmer Arcs, Ann. Sof. Univ. Fac. Math. Inf., 101 (2013), 183-192.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [13] [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. Google Scholar [15] G. Solomon and J. J. Stiffler, Algebraically punctured cyclic codes, Inform. Control, 8 (1965) 170-179.  Google Scholar

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##### References:
 [1] S. Ball, On intersection sets in Desarguesian affine spaces, European J. Combin., 21 (2000), 441-446. doi: 10.1006/eujc.2000.0350.  Google Scholar [2] I. Boukliev, Optimal Linear Codes - Constructions and Bounds, Ph.D thesis, Sofia, 1996. Google Scholar [3] S. Dodunekov and J. Simonis, Codes and projective multisets, Electr. J. Combin., 5 (1998), #R37.  Google Scholar [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.  Google Scholar [5] J. H. Griesmer, A bound for error-correcting codes, IBM J. Res. Develop., 4 (1960), 532-542.  Google Scholar [6] R. Hill, Optimal linear codes, in Cryptography and Coding (ed. C. Mitchell), Oxford Univ. Press, 1992, 75-104.  Google Scholar [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.  Google Scholar [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. Google Scholar [9] I. Landjev and A. Rousseva, On the Extendability of Griesmer Arcs, Ann. Sof. Univ. Fac. Math. Inf., 101 (2013), 183-192.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [13] [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. Google Scholar [15] G. Solomon and J. J. Stiffler, Algebraically punctured cyclic codes, Inform. Control, 8 (1965) 170-179.  Google Scholar
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