Divisible Arcs, Divisible Codes, and the Extension Problem for Arcs and Codes

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).

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