Intersections in Projective Space II: Pencils of Quadrics

On the Largest Caps Contained in the Klein Quadric of PG(5, q), q Odd

“…Assume that the tangent point of a tangent line through r to C is t 11 and assume that t 11 also belongs to C and the set of secant lines through r in rL j to C (1) j . Identifying these secant sets with the corresponding sets in PG(1, q) gives a first set O 1 of size (q&1)Â2 and a second set O 2 of size (q\1)Â2.…”

“…Then C (1) 1 is contained in the intersection of the quadrics of the pencil constructed in Lemma 3.1.…”

“…This implied that r is external to at least two conics C (1) i and at least two conics C (2) j . Let t 1 j and t 2 j be the elements of PG(1, q), or its extension PG(1, q 2 ), corresponding to the tangent lines through r to C (1) j . In Lemma 3.1, a pencil of quadrics, containing rQ + (3, q), and containing at least (q&1)Â14 conics C We first of all show that C…”

“…Then the intersection of H with the cone LE 3, q is a (q 3 +q 2 +q+1)-cap contained in the Klein quadric. In [1], Hirschfeld and Bruen proved that there exist in PG(3, q) three non-isomorphic pencils of quadrics defined by two skew elliptic quadrics in PG (3, q), which can be used to construct these (q 3 +q 2 +q+1)-caps contained in the Klein quadric.…”

Near-MDS codes arising from algebraic curves

“…However, the classical result stating that every elliptic quartic is birationally isomorphic to an elliptic cubic (and vice versa) holds true over any GF(q). For more general results on elliptic quartics, see [4,1]. Here we only mention that an elliptic quartic 4 is the base curve of a pencil of quadrics and that the number N q of GF(q)-rational points of 4 equals the number of GF(q)-rational cones plus twice the number of GF(q)-rational hyperbolic quadrics in the pencil (see [1,Corollary p. 257]).…”