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).

中文翻译：

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可分开的弧，可分开的代码以及弧和代码的扩展问题

在较早的论文中，我们开发了一种统一的方法来解决PG（k -1，q）中弧的可扩展性问题，以及等效地解决有限域上的线性代码的问题。我们定义了称为（t mod q）弧的特殊弧，并证明了给定弧的可扩展性取决于特殊的双弧的结构，结果证明是（t mod q）弧。在本文中，我们研究了（t mod q）弧的一般结构。我们证明每个这样的弧都是超平面的补码之和。此外，我们描述这种电弧为小值吨，其在壳体吨= 2为我们提供了Maruta关于代码可扩展性的定理的另一种证明。这一结果是几何学上等效形状的声明，在PG（每2-quasidivisible弧ķ - 1，q），q ≥5，q奇，是可扩展的。最后，我们介绍了我们的方法在PG（3，q）中上限的可扩展性问题中的应用。