$2$-arcs of maximal size in the affine and the projective Hjelmslev plane over $\mathbb Z$25

Michael Kiermaier; Matthias Koch; Sascha Kurz

doi:10.3934/amc.2011.5.287

Article Contents

2011, Volume 5, Issue 2: 287-301. Doi: 10.3934/amc.2011.5.287

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$2$-arcs of maximal size in the affine and the projective Hjelmslev plane over $\mathbb Z$₂₅

1.
Institut für Mathematik, Universität Bayreuth, D-95440 Bayreuth

Received: March 31, 2010

Revised: July 31, 2010

Published: April 2011

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Abstract

It is shown that the maximal size of a $2$-arc in the projective Hjelmslev plane over $\mathbb Z$₂₅ is $21$, and the $(21,2)$-arc is unique up to isomorphism. Furthermore, all maximal $(20,2)$-arcs in the affine Hjelmslev plane over $\mathbb Z$₂₅ are classified up to isomorphism.

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Mathematics Subject Classification: Primary: 51C05, 51E21; Secondary: 94B05.

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References

[1]	A. Cronheim, Dual numbers, Witt vectors, and Hjelmslev planes, Geom. Dedicata, 7 (1978), 287-302.doi: 10.1007/BF00151527.
[2]	L. Hemme and D. Weijand, Arcs in projektiven Hjelmslev-Ebenen, Fortgeschrittenenpraktikum, Technische Universität München, 1999.
[3]	T. Honold and M. Kiermaier, Classification of maximal arcs in small projective Helmslev geometries, in "Proceedings of the Tenth International Workshop on Algebraic and Combinatorial Coding Theory 2006,'' (2006), 112-117.
[4]	T. Honold and M. Kiermaier, The existence of maximal $(q^2,2)$-arcs in projective Hjelmslev planes over chain rings of odd prime characteristic, in preparation, 2011.
[5]	T. Honold and I. Landjev, Linear codes over finite chain rings, Electr. J. Comb., 7 (2000), #R11.
[6]	T. Honold and I. Landjev, On arcs in projective Hjelmslev planes, Discrete Math., 231 (2001), 265-278.doi: 10.1016/S0012-365X(00)00323-X.
[7]	T. Honold and I. Landjev, On maximal arcs in projective Hjelmslev planes over chain rings of even characteristic, Finite Fields Appl., 11 (2005), 292-304.doi: 10.1016/j.ffa.2004.12.004.
[8]	M. Kiermaier, "Arcs und Codes über endlichen Kettenringen,'' Diploma thesis, Technische Universität München, 2006.
[9]	M. Kiermaier and M. Koch, New complete $2$-arcs in the uniform projective Hjelmslev planes over chain rings of order $25$, in "Proceedings of the Sixth International Workshop on Optimal Codes and Related Topics 2009,'' (2009), 206-113.
[10]	M. Kiermaier and A. Kohnert, Online tables of arcs in projective Hjelmslev planes, http://www.algorithm.uni-bayreuth.de
[11]	M. Kiermaier and A. Kohnert, New arcs in projective Hjelmslev planes over Galois rings, in "Proceedings of the Fifth International Workshop on Optimal Codes and Related Topics 2007,'' (2007), 112-119.
[12]	W. Klingenberg, Projektive und affine Ebenen mit Nachbarelementen, Math. Z., 60 (1954), 384-406.doi: 10.1007/BF01187385.
[13]	D. E. Knuth, Estimating the efficiency of backtrack programs, Math. Comput., 29 (1975), 121-136.doi: 10.2307/2005469.
[14]	A. Kreuzer, "Projektive Hjelmslev-Räume,'' Ph.D. thesis, Technische Universität München, 1988.
[15]	S. Kurz, Caps in $\mathbbZ_n^2$, Serdica J. Comput., 3 (2009), 159-178.
[16]	R. Laue, Construction of combinatorial objects - a tutorial, Bayreuther Math. Schr., 43 (1993), 53-96.
[17]	R. Laue, Constructing objects up to isomorphism, simple $9$-designs with small parameters, in "Algebraic Combinatorics and Applications,'' Springer, Berlin, (2001), 232-260.
[18]	H. Lüneburg, Affine Hjelmslev-Ebenen mit transitiver Translationsgruppe, Math. Z., 79 (1962), 260-288.doi: 10.1007/BF01193123.
[19]	F. Margot, Pruning by isomorphism in branch-and-cut, Math. Programming, 94 (2002), 71-90.doi: 10.1007/s10107-002-0358-2.
[20]	B. McKay, Nauty, Version 2.2, http://cs.anu.edu.au/~bdm/nauty/
[21]	A. A. Nečaev, Finite principal ideal rings, Math. USSR-Sb., 20 (1973), 364-382.doi: 10.1070/SM1973v020n03ABEH001880.
[22]	A. A. Nechaev, Finite rings with applications, in "Handbook of Algebra'' (ed. M. Hazewinkel), Elsevier Science Publishers, (2008), 213-320.
[23]	R. Raghavendran, Finite associative rings, Composito Math., 21 (1969), 195-229.
[24]	R. C. Read, Every one a winner or how to avoid isomorphism search when cataloguing combinatorial configurations, Ann. Discrete Math., 2 (1978), 107-120.doi: 10.1016/S0167-5060(08)70325-X.
[25]	B. Schmalz, The $t$-designs with prescribed automorphism group, new simple $6$-designs, J. Comb. Des., 1 (1993), 125-170.doi: 10.1002/jcd.3180010204.