Weighted \((k,n)\)-arcs of Type \((n-q,n)\) and Maximum Size of \((h,m)\)-arcs in \(\mathrm{PG}(2,q)\)

Mustafa T. Yaseen, Ali Hasan Ali, Ibrahim A. Shanan

Abstract


In this paper, we introduce a generalized weighted \((k, n)\)-arc of two types in the projective plane of order \(q\), where \(q\) is an odd prime number. The sided result of this work is finding the largest size of a complete \((h, m)\)-arcs in \(\mathrm{PG}(2, q)\), where \(h\) represents a point of weight zero of a weighted \((k, n)\)-arc. Also, we prove that a \(\big(\frac{q(q-1)}{2}+1, \frac{q+1}{2}\big)\)-arc is a maximal arc in \(\mathrm{PG}(2, q)\).

Keywords


\((k, n)\)-arcs; Weighted \((k,n)\)-arc; \(\mathrm{PG}(2, q)\); \(\mathrm{PG}(2, \text{prime})\); Projective plane; Galois plane; Algebraic geometry

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DOI: http://dx.doi.org/10.26713%2Fcma.v10i3.1275

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