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The existence of maximal (q², 2)-arcs in projective Hjelmslev planes over chain rings of length 2 and odd prime characteristic

Title data

Honold, Thomas ; Kiermaier, Michael:
The existence of maximal (q², 2)-arcs in projective Hjelmslev planes over chain rings of length 2 and odd prime characteristic.
In: Designs, Codes and Cryptography. Vol. 68 (2013) Issue 1–3 . - pp. 105-126.
ISSN 1573-7586
DOI: https://doi.org/10.1007/s10623-012-9653-y

Project information

Project financing: Deutsche Forschungsgemeinschaft
National Natural Science Foundation of China (Grant No. 60872063) Chinese Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 200803351027) Deutsche Forschungsgemeinschaft (Grant No. WA 1666/4-1)

Abstract in another language

We prove that (q², 2)-arcs exist in the projective Hjelmslev plane PHG(2,R) over a chain ring R of length 2, order |R|=q² and prime characteristic. For odd prime characteristic, our construction solves the maximal arc problem. For characteristic 2, an extension of the above construction yields the lower bound q²+2 on the maximum size of a 2-arc in PHG(2,R). Translating the arcs into codes, we get linear [q³,6,q³-q²-q] codes over F_q for every prime power q>1 and linear [q³+q,6,q³-q²-1] codes over F_q for the special case q=2ʳ. Furthermore, we construct 2-arcs of size (q+1)²/4 in the planes PHG(2,R) over Galois rings R of length 2 and odd characteristic p².

Further data

Item Type: Article in a journal
Refereed: Yes
Institutions of the University: Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II (Computer Algebra)
Faculties
Faculties > Faculty of Mathematics, Physics und Computer Science
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics
Result of work at the UBT: Yes
DDC Subjects: 500 Science > 510 Mathematics
Date Deposited: 20 Nov 2014 07:55
Last Modified: 23 Nov 2022 07:37
URI: https://eref.uni-bayreuth.de/id/eprint/3716