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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

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), 287-302. doi: 10.1007/BF00151527.  Google Scholar

[2]

L. Hemme and D. Weijand, Arcs in projektiven Hjelmslev-Ebenen, Fortgeschrittenenpraktikum, Technische Universität München, 1999. Google Scholar

[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. 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), #R11.  Google Scholar

[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.  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), 292-304. doi: 10.1016/j.ffa.2004.12.004.  Google Scholar

[8]

M. Kiermaier, "Arcs und Codes über endlichen Kettenringen,'' Diploma thesis, Technische Universität München, 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 "Proceedings of the Sixth International Workshop on Optimal Codes and Related Topics 2009,'' (2009), 206-113. 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 "Proceedings of the Fifth International Workshop on Optimal Codes and Related Topics 2007,'' (2007), 112-119. Google Scholar

[12]

W. Klingenberg, Projektive und affine Ebenen mit Nachbarelementen, Math. Z., 60 (1954), 384-406. doi: 10.1007/BF01187385.  Google Scholar

[13]

D. E. Knuth, Estimating the efficiency of backtrack programs, Math. Comput., 29 (1975), 121-136. doi: 10.2307/2005469.  Google Scholar

[14]

A. Kreuzer, "Projektive Hjelmslev-Räume,'' Ph.D. thesis, Technische Universität München, 1988.  Google Scholar

[15]

S. Kurz, Caps in $\mathbbZ_n^2$, Serdica J. Comput., 3 (2009), 159-178.  Google Scholar

[16]

R. Laue, Construction of combinatorial objects - a tutorial, Bayreuther Math. Schr., 43 (1993), 53-96.  Google Scholar

[17]

R. Laue, Constructing objects up to isomorphism, simple $9$-designs with small parameters, in "Algebraic Combinatorics and Applications,'' Springer, Berlin, (2001), 232-260.  Google Scholar

[18]

H. Lüneburg, Affine Hjelmslev-Ebenen mit transitiver Translationsgruppe, Math. Z., 79 (1962), 260-288. doi: 10.1007/BF01193123.  Google Scholar

[19]

F. Margot, Pruning by isomorphism in branch-and-cut, Math. Programming, 94 (2002), 71-90. 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), 364-382. doi: 10.1070/SM1973v020n03ABEH001880.  Google Scholar

[22]

A. A. Nechaev, Finite rings with applications, in "Handbook of Algebra'' (ed. M. Hazewinkel), Elsevier Science Publishers, (2008), 213-320.  Google Scholar

[23]

R. Raghavendran, Finite associative rings, Composito Math., 21 (1969), 195-229.  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), 107-120. 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), 125-170. doi: 10.1002/jcd.3180010204.  Google Scholar

show all references

References:
[1]

A. Cronheim, Dual numbers, Witt vectors, and Hjelmslev planes, Geom. Dedicata, 7 (1978), 287-302. doi: 10.1007/BF00151527.  Google Scholar

[2]

L. Hemme and D. Weijand, Arcs in projektiven Hjelmslev-Ebenen, Fortgeschrittenenpraktikum, Technische Universität München, 1999. Google Scholar

[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. 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), #R11.  Google Scholar

[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.  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), 292-304. doi: 10.1016/j.ffa.2004.12.004.  Google Scholar

[8]

M. Kiermaier, "Arcs und Codes über endlichen Kettenringen,'' Diploma thesis, Technische Universität München, 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 "Proceedings of the Sixth International Workshop on Optimal Codes and Related Topics 2009,'' (2009), 206-113. 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 "Proceedings of the Fifth International Workshop on Optimal Codes and Related Topics 2007,'' (2007), 112-119. Google Scholar

[12]

W. Klingenberg, Projektive und affine Ebenen mit Nachbarelementen, Math. Z., 60 (1954), 384-406. doi: 10.1007/BF01187385.  Google Scholar

[13]

D. E. Knuth, Estimating the efficiency of backtrack programs, Math. Comput., 29 (1975), 121-136. doi: 10.2307/2005469.  Google Scholar

[14]

A. Kreuzer, "Projektive Hjelmslev-Räume,'' Ph.D. thesis, Technische Universität München, 1988.  Google Scholar

[15]

S. Kurz, Caps in $\mathbbZ_n^2$, Serdica J. Comput., 3 (2009), 159-178.  Google Scholar

[16]

R. Laue, Construction of combinatorial objects - a tutorial, Bayreuther Math. Schr., 43 (1993), 53-96.  Google Scholar

[17]

R. Laue, Constructing objects up to isomorphism, simple $9$-designs with small parameters, in "Algebraic Combinatorics and Applications,'' Springer, Berlin, (2001), 232-260.  Google Scholar

[18]

H. Lüneburg, Affine Hjelmslev-Ebenen mit transitiver Translationsgruppe, Math. Z., 79 (1962), 260-288. doi: 10.1007/BF01193123.  Google Scholar

[19]

F. Margot, Pruning by isomorphism in branch-and-cut, Math. Programming, 94 (2002), 71-90. 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), 364-382. doi: 10.1070/SM1973v020n03ABEH001880.  Google Scholar

[22]

A. A. Nechaev, Finite rings with applications, in "Handbook of Algebra'' (ed. M. Hazewinkel), Elsevier Science Publishers, (2008), 213-320.  Google Scholar

[23]

R. Raghavendran, Finite associative rings, Composito Math., 21 (1969), 195-229.  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), 107-120. 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), 125-170. doi: 10.1002/jcd.3180010204.  Google Scholar

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