On multiple caps in finite projective spaces. (English) Zbl 1204.51016
Let \(PG(t, q)\) be the \(t\)-dimensional projective geometry over the finite field with \(q\) elements. A \((k, n)\)-cap in \(PG(t, q)\) is a set of \(k\) points, some \(n\), but no \(n + 1\) of which are collinear. The authors consider new results on \((k, n)\)-caps with \(n > 2\). Here the authors provide a lower bound on the size of such caps. Furthermore, they generalize two product constructions for \((k, 2)\)-caps to caps with larger \(n\). They give explicit constructions for good caps with small \(n\). In particular, they determine the largest size of a \((k, 3)\)-cap in \(PG(3, 5)\), which turns out to be 44. The results on caps in \(PG(3, 5)\) provide a solution to four of the eight open instances of the main coding theory problem for \(q = 5\) and \(k = 4\).
Reviewer: Giorgio Faina (Perugia)
MSC:
| 51E22 | Linear codes and caps in Galois spaces |
| 94B05 | Linear codes (general theory) |
| 94B65 | Bounds on codes |
Keywords:
caps; multiple caps; linear codes; Griesmer bound; Griesmer codes; finite projective geometriesReferences:
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