A new lower bound for the smallest complete (k, n)-arc in $$\mathrm {PG}(2,q)$$ PG ( 2 , q )
Salam Abdulqader Falih Alabdullah, James W. P. Hirschfeld · Designs Codes and Cryptography · 2018
In $$\mathrm {PG}(2,q)$$ , the projective plane over the field $$\mathbf{F}_{q}$$ of q elements, a (k, n)-arc is a set $$\mathcal {K}$$ of k points with at most n points on any line of the plane. A fundamental question is to determine the values of k for which $$\mathcal {K}$$ is complete, that is, not contained in a $$(k+1,n)$$ -arc. In particular, what are the smallest and largest values of k for a complete $$\mathcal {K}$$ , denoted by $$t_n(2,q)$$ and $$m_n(2,q)$$ ? Here, a new lower bound for $$t_n(2,q)$$ is established and compared to known values for small q.