2-arcs of maximal size in the affine and the projective Hjelmslev plane over \mathbb Z<SUB><I>25</I></SUB>

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May 2011, 5(2): 287-301. doi: 10.3934/amc.2011.5.287

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

Michael Kiermaier ^1, , Matthias Koch ^1, and Sascha Kurz ^1,

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

Received April 2010 Revised August 2010 Published May 2011

Abstract
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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.

Keywords: Galois ring., Hjelmslev geometry, nite chain ring, arc.

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

Citation: Michael Kiermaier, Matthias Koch, Sascha Kurz. $2$-arcs of maximal size in the affine and the projective Hjelmslev plane over $\mathbb Z$₂₅. Advances in Mathematics of Communications, 2011, 5 (2) : 287-301. doi: 10.3934/amc.2011.5.287

References:

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

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References:

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

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