Divisible Arcs, Divisible Codes, and the Extension Problem for Arcs and Codes
Ivan Landjev, Assia Rousseva · Problems of Information Transmission · 2019
In an earlier paper we developed a unified approach to the extendability problem for arcs in PG( k - 1, q ) and, equivalently, for linear codes over finite fields. We defined a special class of arcs called ( t mod q )-arcs and proved that the extendabilty of a given arc depends on the structure of a special dual arc, which turns out to be a ( t mod q )-arc. In this paper, we investigate the general structure of ( t mod q )-arcs. We prove that every such arc is a sum of complements of hyperplanes. Furthermore, we characterize such arcs for small values of t , which in the case t = 2 gives us an alternative proof of the theorem by Maruta on the extendability of codes. This result is geometrically equivalent to the statement that every 2-quasidivisible arc in PG( k - 1, q ), q ≥ 5, q odd, is extendable. Finally, we present an application of our approach to the extendability problem for caps in PG(3, q ).