The existence of maximal (q
2, 2)-arcs in projective Hjelmslev planes over chain rings of length 2 and odd prime characteristic |
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Authors: | Thomas Honold Michael Kiermaier |
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Institution: | 1. Department of Information Science and Electronics Engineering, Zhejiang University, 38 Zheda Road, Hangzhou, 310027, China 2. Mathematisches Institut, Universit?t Bayreuth, 95440, Bayreuth, Germany
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Abstract: | We prove that (q 2, 2)-arcs exist in the projective Hjelmslev plane PHG(2, R) over a chain ring R of length 2, order |R| = q 2 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 q 2 + 2 on the maximum size of a 2-arc in PHG(2, R). Translating the arcs into codes, we get linear q 3, 6, q 3 ?q 2 ?q] codes over ${\mathbb {F}_q}$ for every prime power q > 1 and linear q 3 + q, 6,q 3 ?q 2 ?1] codes over ${\mathbb {F}_q}$ for the special case q = 2 r . 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 p 2. |
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