On the (22, 4)-arcs in PG(2, 7) and related codes
An \((n,t)_q\)-arc \(K\) in \(PG(2,q)\) is a set of \(n\) points such that every line of \(PG(2,q)\) meets \(K\) in at most \(t\) points. The largest size \(n\) for an \((n,4)_7\)-arc is 22. The authors classify all \((22,4)_7\)-arcs. They prove that there are exactly three inequivalent \((22,4)_7\)-arcs. These three \((22,4)_7\)-arcs are members of three infinite classes of \(((q^2-q+2)/2,(q+1)/2)_q\)-arcs for \(q\) odd. The infinite classes are constructed by using properties of the internal points of a conic in \(PG(2,q)\), \(q\) odd. The classification of the \((22,4)_7\)-arcs is then used to prove the non-existence of \([23,4,18]_7\)-codes, showing that 24 is the shortest length for an \([n,4,18]_7\)-code.
scientific article; zbMATH DE number 1924770