Maximal [formula omitted]-arcs in [formula omitted

Abstract

In this paper we show, using a computer-based search exploiting relations of inclusion between arcs and ( n , 3 ) -arcs and projective equivalence properties, that the largest size of a complete ( n , 3 ) -arc in PG ( 2 , 13 ) is 23 and that only seven non-equivalent ( 23 , 3 ) -arcs exist. From this result, we deduce the non-existence of some [ n , k , n - k ] 13 linear codes and bounds on the minimum distance of some [ n , 3 , d ] 13 linear codes. Moreover, we determine the spectrum of the sizes of the complete ( n , 3 ) -arcs in PG ( 2 , 13 ) and the classification of the smallest complete ( n , 3 ) -arcs.