February  2011, 5(1): 11-21. doi: 10.3934/amc.2011.5.11

The dual construction for arcs in projective Hjelmslev spaces

1. 

Zhejiang Provincial Key Laboratory of Information Network Technology, and Department of Information and Electronic Engineering, Zhejiang University, 38 Zheda Road, Hangzhou, Zhejiang 310027, China

2. 

New Bulgarian University, 18 Montevideo str., Sofia 1618, Bulgaria, and Institute of Mathematics and Informatics, 8 Acad. G. Bonchev str., Sofia 1113, Bulgaria

Received  February 2010 Revised  September 2010 Published  February 2011

In this paper, we present a duality construction for multiarcs in projective Hjelmslev geometries over chain rings of nilpotency index 2. We compute the parameters of the resulting arcs and discuss some examples.
Citation: Thomas Honold, Ivan Landjev. The dual construction for arcs in projective Hjelmslev spaces. Advances in Mathematics of Communications, 2011, 5 (1) : 11-21. doi: 10.3934/amc.2011.5.11
References:
[1]

A. Brouwer and M. van Eupen, The correspondence between projective codes and 2-weight codes, Des. Codes Crypt., 11 (1997), 262-266. doi: 10.1023/A:1008294128110.

[2]

E. Byrne, M. Greferath and T. Honold, Ring geometries, two-weight codes and strongly regular graphs, Des. Codes Crypt., 48 (2008), 1-16. doi: 10.1007/s10623-007-9136-8.

[3]

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

[4]

S. Dodunekov and J. Simonis, Codes and projective multisets, Electr. J. Combin., 5 (1998), 23.

[5]

J. W. P. Hirschfeld, "Projective Geometries over Finite Fields,'' $2^{nd}$ edition, Oxford University Press, 1998.

[6]

T. Honold, M. Kiermaier and I. Landjev, New arcs of maximal size in projective Hjelmslev planes of order nine, Comptes Rendus l'Acad. Bulgare Sci., 63 (2010), 171-180.

[7]

T. Honold and I. Landjev, Projective Hjelmslev geometries, in "Proceedings of the International Workshop on Optimal Codes,'' Sozopol, Bulgaria, (1998), 97-115.

[8]

T. Honold and I. Landjev, Linearly representable codes over chain rings, Abhandlungen math. Seminar Univ. Hamburg, 69 (1999), 187-203. doi: 10.1007/BF02940872.

[9]

T. Honold and I. Landjev, Linear codes over finite chain rings, Electr. J. Combin., 7 (2000), 22.

[10]

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.

[11]

T. Honold and I. Landjev, On maximal arcs in projective Hjelmslev planes, Finite Fields Appl., 11 (2005), 292-304. doi: 10.1016/j.ffa.2004.12.004.

[12]

M. Kiermaier and A. Kohnert, New arcs in projective Hjelmslev planes over Galois rings, in "Optimal Codes and Related Topics,'' White Lagoon, Bulgaria, (2007), 112-119.

[13]

A. Kreuzer, Fundamental theorem of projective Hjelmslev spaces, Mitteilungen Math. Gesellschaft Hamburg, 12 (1991), 809-817.

[14]

I. Landjev and T. Honold, Arcs in projective Hjelmslev planes, Discrete Math. Appl., 11 (2001), 53-70. doi: 10.1515/dma.2001.11.1.53.

[15]

A. A. Nechaev, Finite principal ideal rings, Sbornik. Math., 20 (1973), 364-382. doi: 10.1070/SM1973v020n03ABEH001880.

[16]

R. Raghavendran, Finite associative rings, Compositio Math., 21 (1969), 195-229.

show all references

References:
[1]

A. Brouwer and M. van Eupen, The correspondence between projective codes and 2-weight codes, Des. Codes Crypt., 11 (1997), 262-266. doi: 10.1023/A:1008294128110.

[2]

E. Byrne, M. Greferath and T. Honold, Ring geometries, two-weight codes and strongly regular graphs, Des. Codes Crypt., 48 (2008), 1-16. doi: 10.1007/s10623-007-9136-8.

[3]

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

[4]

S. Dodunekov and J. Simonis, Codes and projective multisets, Electr. J. Combin., 5 (1998), 23.

[5]

J. W. P. Hirschfeld, "Projective Geometries over Finite Fields,'' $2^{nd}$ edition, Oxford University Press, 1998.

[6]

T. Honold, M. Kiermaier and I. Landjev, New arcs of maximal size in projective Hjelmslev planes of order nine, Comptes Rendus l'Acad. Bulgare Sci., 63 (2010), 171-180.

[7]

T. Honold and I. Landjev, Projective Hjelmslev geometries, in "Proceedings of the International Workshop on Optimal Codes,'' Sozopol, Bulgaria, (1998), 97-115.

[8]

T. Honold and I. Landjev, Linearly representable codes over chain rings, Abhandlungen math. Seminar Univ. Hamburg, 69 (1999), 187-203. doi: 10.1007/BF02940872.

[9]

T. Honold and I. Landjev, Linear codes over finite chain rings, Electr. J. Combin., 7 (2000), 22.

[10]

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.

[11]

T. Honold and I. Landjev, On maximal arcs in projective Hjelmslev planes, Finite Fields Appl., 11 (2005), 292-304. doi: 10.1016/j.ffa.2004.12.004.

[12]

M. Kiermaier and A. Kohnert, New arcs in projective Hjelmslev planes over Galois rings, in "Optimal Codes and Related Topics,'' White Lagoon, Bulgaria, (2007), 112-119.

[13]

A. Kreuzer, Fundamental theorem of projective Hjelmslev spaces, Mitteilungen Math. Gesellschaft Hamburg, 12 (1991), 809-817.

[14]

I. Landjev and T. Honold, Arcs in projective Hjelmslev planes, Discrete Math. Appl., 11 (2001), 53-70. doi: 10.1515/dma.2001.11.1.53.

[15]

A. A. Nechaev, Finite principal ideal rings, Sbornik. Math., 20 (1973), 364-382. doi: 10.1070/SM1973v020n03ABEH001880.

[16]

R. Raghavendran, Finite associative rings, Compositio Math., 21 (1969), 195-229.

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