Abstract:
The (k,n)-arcs in projective Hjelmslev plane PHG(R3R) over a finite chain ring R are considered. We prove general upper bounds on the cardinality of such arcs and establish the maximum
possible size of the projective (k,n)-arcs with n∈{q2,…,q2+q−1}. Constructions of projective
arcs in the Hjelmslev planes over the chain rings with 4 and 9 elements are also given.
This publication is cited in the following 15 articles:
Honold T., Kiermaier M., Landjev I., “New Upper Bounds on the Maximal Size of An Arc in a Projective Hjelmslev Plane”, Proceedings of the 2020 Seventeenth International Workshop on Algebraic and Combinatorial Coding Theory Algebraic and Combinatorial Coding Theory (Acct 2020): Proceedings of the Seventeenth International Workshop on Algebraic and Combinatorial Coding Theory Acct 2020, IEEE, 2020, 67–76
Rostami E., Nekooei R., “Computation of Minimum Hamming Weight For Linear Codes”, Iran. J. Math. Sci. Inform., 14:1 (2019), 81–93
Honold T., Landjev I., “Non-Free Extensions of the Simplex Codes Over a Chain Ring with Four Elements”, Des. Codes Cryptogr., 66:1-3, SI (2013), 27–38
Honold T., Kiermaier M., “The maximal size of 6- and 7-arcs in projective Hjelmslev planes over chain rings of order 9”, Science China-Mathematics, 55:1 (2012), 73–92
Honold T., Landjev I., “The Dual Construction for Arcs in Projective Hjelmslev Spaces”, Adv Math Commun, 5:1 (2011), 11–21
Landjev I., Rousseva A., “Characterization of Some Optimal Arcs”, Adv Math Commun, 5:2 (2011), 317–331
Boev S., Honold T., Landjev I., “Optimal Arcs in Hjelmslev Spaces of Higher Dimension”, C R Acad Bulgare Sci, 64:5 (2011), 625–632
Honold T., Kiermaier M., Landjev I., “New arcs of maximal size in projective hjelmslev planes of order nine”, Comptes-rendus de l'Académie Bulgare des Sciences, 63:2 (2010), 171–180
Kohnert A., “Sets of Type (d(1), d(2)) in Projective Hjelmslev Planes over Galois Rings”, Algorithmic Algebraic Combinatorics and Grobner Bases, 2009, 269–278
Honold T., Landjev I., “Linear Codes over Finite Chain Rings and Projective Hjelmslev Geometries”, Codes Over Rings, Series on Coding Theory and Cryptology, 6, 2009, 60–123
Landjev I., “On blocking sets in projective Hjelmslev planes”, Adv. Math. Commun., 1:1 (2007), 65–81
Ward H.N., “Arcs, minihypers, and the classification of three-dimensional Griesmer codes”, Advances in Coding Theory and Cryptography, Series on Coding Theory and Cryptology, 3, 2007, 33–50
Honold T., Landjev I., “On maximal arcs in projective Hjelmslev planes over chain rings of even characteristic”, Finite Fields Appl., 11:2 (2005), 292–304
Landjev I., Rousseva A., Maruta T., Hill R., “On optimal codes over the field with five elements”, Des. Codes Cryptogr., 29:1-3 (2003), 165–175
Shiromoto K., Storme L., “A Griesmer bound for linear codes over finite quasi-Frobenius rings”, International Workshop on Coding and Cryptography (WCC 2001) (Paris), Discrete Appl. Math., 128, no. 1, 2003, 263–274
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