Divisible Arcs, Divisible Codes, and the Extension Problem for Arcs and Codes
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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).
Key wordsfinite projective geometries arcs blocking sets divisible arcs quasidivisible arcs, Griesmer bound (t mod q)-arcs extendable arcs minihypers caps
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This research has been supported by the Science Research Fund of Sofia University under Contract no. 80-10-81/15.04.2019.
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