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

Titelangaben

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. Bd. 68 (Juli 2013) Heft 1–3 . - S. 105-126.
ISSN 0925-1022
DOI: https://doi.org/10.1007/s10623-012-9653-y

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Abstract

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.