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A $\mathbb Z$4-linear code of high minimum Lee distance derived from a hyperoval
$2$-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 |
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. |
show all references
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. |
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