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

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.
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

[1]

Claude Carlet, Juan Carlos Ku-Cauich, Horacio Tapia-Recillas. Bent functions on a Galois ring and systematic authentication codes. Advances in Mathematics of Communications, 2012, 6 (2) : 249-258. doi: 10.3934/amc.2012.6.249

[2]

Alexandre Fotue-Tabue, Edgar Martínez-Moro, J. Thomas Blackford. On polycyclic codes over a finite chain ring. Advances in Mathematics of Communications, 2020, 14 (3) : 455-466. doi: 10.3934/amc.2020028

[3]

Liqin Qian, Xiwang Cao. Character sums over a non-chain ring and their applications. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020134

[4]

Stefanella Boatto. Curvature perturbations and stability of a ring of vortices. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 349-375. doi: 10.3934/dcdsb.2008.10.349

[5]

Heide Gluesing-Luerssen, Fai-Lung Tsang. A matrix ring description for cyclic convolutional codes. Advances in Mathematics of Communications, 2008, 2 (1) : 55-81. doi: 10.3934/amc.2008.2.55

[6]

Ke Gu, Xinying Dong, Linyu Wang. Efficient traceable ring signature scheme without pairings. Advances in Mathematics of Communications, 2020, 14 (2) : 207-232. doi: 10.3934/amc.2020016

[7]

Tomasz Cieślak. Trudinger-Moser type inequality for radially symmetric functions in a ring and applications to Keller-Segel in a ring. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2505-2512. doi: 10.3934/dcdsb.2013.18.2505

[8]

Carlos Garca-Azpeitia, Jorge Ize. Bifurcation of periodic solutions from a ring configuration of discrete nonlinear oscillators. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 975-983. doi: 10.3934/dcdss.2013.6.975

[9]

Satoshi Kosugi, Yoshihisa Morita. Phase pattern in a Ginzburg-Landau model with a discontinuous coefficient in a ring. Discrete & Continuous Dynamical Systems, 2006, 14 (1) : 149-168. doi: 10.3934/dcds.2006.14.149

[10]

R. Yamapi, R.S. MacKay. Stability of synchronization in a shift-invariant ring of mutually coupled oscillators. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 973-996. doi: 10.3934/dcdsb.2008.10.973

[11]

Steven T. Dougherty, Cristina Fernández-Córdoba, Roger Ten-Valls, Bahattin Yildiz. Quaternary group ring codes: Ranks, kernels and self-dual codes. Advances in Mathematics of Communications, 2020, 14 (2) : 319-332. doi: 10.3934/amc.2020023

[12]

Joan-Josep Climent, Juan Antonio López-Ramos. Public key protocols over the ring $E_{p}^{(m)}$. Advances in Mathematics of Communications, 2016, 10 (4) : 861-870. doi: 10.3934/amc.2016046

[13]

Steven T. Dougherty, Joe Gildea, Abidin Kaya, Bahattin Yildiz. New self-dual and formally self-dual codes from group ring constructions. Advances in Mathematics of Communications, 2020, 14 (1) : 11-22. doi: 10.3934/amc.2020002

[14]

Joseph G. Conlon, André Schlichting. A non-local problem for the Fokker-Planck equation related to the Becker-Döring model. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 1821-1889. doi: 10.3934/dcds.2019079

[15]

Habibul Islam, Om Prakash, Ram Krishna Verma. New quantum codes from constacyclic codes over the ring $ R_{k,m} $. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020097

[16]

Denis Blackmore, Jyoti Champanerkar, Chengwen Wang. A generalized Poincaré-Birkhoff theorem with applications to coaxial vortex ring motion. Discrete & Continuous Dynamical Systems - B, 2005, 5 (1) : 15-33. doi: 10.3934/dcdsb.2005.5.15

[17]

Xiaoxue Zhao, Zhuchun Li, Xiaoping Xue. Formation, stability and basin of phase-locking for Kuramoto oscillators bidirectionally coupled in a ring. Networks & Heterogeneous Media, 2018, 13 (2) : 323-337. doi: 10.3934/nhm.2018014

[18]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge : A study of fractional calculus on metric graph. Networks & Heterogeneous Media, 2021, 16 (2) : 155-185. doi: 10.3934/nhm.2021003

[19]

Lin Yi, Xiangyong Zeng, Zhimin Sun. On finite length nonbinary sequences with large nonlinear complexity over the residue ring $ \mathbb{Z}_{m} $. Advances in Mathematics of Communications, 2021, 15 (4) : 701-720. doi: 10.3934/amc.2020091

[20]

Ivan Landjev. On blocking sets in projective Hjelmslev planes. Advances in Mathematics of Communications, 2007, 1 (1) : 65-81. doi: 10.3934/amc.2007.1.65

2020 Impact Factor: 0.935

Metrics

  • PDF downloads (60)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]