We construct a linear binary [20,6,8]-code using a complete cap of PG(5,2) obtained extending a c... more We construct a linear binary [20,6,8]-code using a complete cap of PG(5,2) obtained extending a complete cap of the Klein quadric.
ABSTRACT Infinite families of (q+1)(q+1)-ovoids and (q2+1)(q2+1)-tight sets of the symplectic pol... more ABSTRACT Infinite families of (q+1)(q+1)-ovoids and (q2+1)(q2+1)-tight sets of the symplectic polar space W(5,q)W(5,q), q even, are constructed. The (q+1)(q+1)-ovoids arise from relative hemisystems of the Hermitian surface H(3,q2)H(3,q2) and from certain orbits of the Suzuki group Sz(q)Sz(q) in his projective 4-dimensional representation. The tight sets are closely related to the geometry of an ovoid of W(3,q)W(3,q). Other constructions of sporadic intriguing sets are also given.
ABSTRACT Several infinite families of (0,α)(0,α)-sets, α≥1α≥1, of finite classical and non-classi... more ABSTRACT Several infinite families of (0,α)(0,α)-sets, α≥1α≥1, of finite classical and non-classical generalized quadrangles are constructed. When α=1α=1 a (0,α)(0,α)-set of a generalized quadrangle is a partial ovoid. We construct a maximal partial ovoid of H(4,q2)H(4,q2), for any q , of size 2q3+q2+12q3+q2+1, which generalizes the unique largest partial ovoid of H(4,4)H(4,4) of size 21 found in [11], and a maximal partial ovoid of Q−(5,q)Q−(5,q) of size (q+1)2(q+1)2, for any q . A tight set of a GQ(q−1,q+1)GQ(q−1,q+1) is also provided.
An ovoid of PG(3, q), q > 2, is a set of q, + 1 points of PG(3, q), no three of wh... more An ovoid of PG(3, q), q > 2, is a set of q, + 1 points of PG(3, q), no three of which are collinear. The only known ovoids of PG(3, q) are the elliptic quadrics, which exist for all q, and the Suzuki-Tits ovoids, which exist for q = 2,, e = 3 odd, [10]. It is well known
... The matrix associated to the Gale transform E of E is ... transitively on the plane sections ... more ... The matrix associated to the Gale transform E of E is ... transitively on the plane sections of E. S is the so-called Witt design W10; The isomorphism group of S is denoted by M10 and is isomorphic to a proper subgroup of PGL(2, 9) containing PSL(2, 9). G in its 6-dimensional ...
We classify, up to isomorphism, maximal curves covered by the Hermitian curve \mathcal H by a pri... more We classify, up to isomorphism, maximal curves covered by the Hermitian curve \mathcal H by a prime degree Galois covering. We also compute the genus of maximal curves obtained by the quotient of \mathcal H by several automorphisms groups. Finally we discuss the value for the third largest genus that a maximal curve can have.
Certain affine sets arising from spreads of the projective space PG(3, q) are investigated. The a... more Certain affine sets arising from spreads of the projective space PG(3, q) are investigated. The affine set arising from a Lüneburg spread is studied in detail.
We describe a simple method to construct many different linear codes arising from caps in a proje... more We describe a simple method to construct many different linear codes arising from caps in a projective space whose automorphism group is known in advance. This procedure begins with a fairly small point set in an appropriate finite projective space.
The chords of a twisted cubic in PG(3,q) are mapped via their Plücker coordinates to the points o... more The chords of a twisted cubic in PG(3,q) are mapped via their Plücker coordinates to the points of a Veronese surface lying on the Klein quadric in PG(5,q). This correspondence over a finite field gives a cap in PG(5,q), that is, a set of points no three of which are collinear. The dual structure, namely the axes of the osculating developable, is also mapped to a Veronese surface. The two surfaces can be combined to give a larger cap. The constructions can be extended to the chords and axes of an arbitrary (q+1)-arc in PG(3,q) when q is even. An alternative construction for the cap associated to a twisted cubic is given for q odd. [For part I and II see A. A. Bruen and J. W. P. Hirschfeld, Geom. Dedicata 6, 495-509 (1977; Zbl 0394.51005) and Geom. Dedicata 7, 333-353 (1978; Zbl 0394.51006)].
The genus of a maximal curve over a finite field with r^2 elements is either g_0=r(r-1)/2 or less... more The genus of a maximal curve over a finite field with r^2 elements is either g_0=r(r-1)/2 or less than or equal to g_1=(r-1)^2/4. Maximal curves with genus g_0 or g_1 have been characterized up to isomorphism. A natural genus to be studied is g_2=(r-1)(r-3)/8, and for this genus there are two non-isomorphism maximal curves known when r \equiv 3 (mod
We construct a linear binary [20,6,8]-code using a complete cap of PG(5,2) obtained extending a c... more We construct a linear binary [20,6,8]-code using a complete cap of PG(5,2) obtained extending a complete cap of the Klein quadric.
ABSTRACT Infinite families of (q+1)(q+1)-ovoids and (q2+1)(q2+1)-tight sets of the symplectic pol... more ABSTRACT Infinite families of (q+1)(q+1)-ovoids and (q2+1)(q2+1)-tight sets of the symplectic polar space W(5,q)W(5,q), q even, are constructed. The (q+1)(q+1)-ovoids arise from relative hemisystems of the Hermitian surface H(3,q2)H(3,q2) and from certain orbits of the Suzuki group Sz(q)Sz(q) in his projective 4-dimensional representation. The tight sets are closely related to the geometry of an ovoid of W(3,q)W(3,q). Other constructions of sporadic intriguing sets are also given.
ABSTRACT Several infinite families of (0,α)(0,α)-sets, α≥1α≥1, of finite classical and non-classi... more ABSTRACT Several infinite families of (0,α)(0,α)-sets, α≥1α≥1, of finite classical and non-classical generalized quadrangles are constructed. When α=1α=1 a (0,α)(0,α)-set of a generalized quadrangle is a partial ovoid. We construct a maximal partial ovoid of H(4,q2)H(4,q2), for any q , of size 2q3+q2+12q3+q2+1, which generalizes the unique largest partial ovoid of H(4,4)H(4,4) of size 21 found in [11], and a maximal partial ovoid of Q−(5,q)Q−(5,q) of size (q+1)2(q+1)2, for any q . A tight set of a GQ(q−1,q+1)GQ(q−1,q+1) is also provided.
An ovoid of PG(3, q), q > 2, is a set of q, + 1 points of PG(3, q), no three of wh... more An ovoid of PG(3, q), q > 2, is a set of q, + 1 points of PG(3, q), no three of which are collinear. The only known ovoids of PG(3, q) are the elliptic quadrics, which exist for all q, and the Suzuki-Tits ovoids, which exist for q = 2,, e = 3 odd, [10]. It is well known
... The matrix associated to the Gale transform E of E is ... transitively on the plane sections ... more ... The matrix associated to the Gale transform E of E is ... transitively on the plane sections of E. S is the so-called Witt design W10; The isomorphism group of S is denoted by M10 and is isomorphic to a proper subgroup of PGL(2, 9) containing PSL(2, 9). G in its 6-dimensional ...
We classify, up to isomorphism, maximal curves covered by the Hermitian curve \mathcal H by a pri... more We classify, up to isomorphism, maximal curves covered by the Hermitian curve \mathcal H by a prime degree Galois covering. We also compute the genus of maximal curves obtained by the quotient of \mathcal H by several automorphisms groups. Finally we discuss the value for the third largest genus that a maximal curve can have.
Certain affine sets arising from spreads of the projective space PG(3, q) are investigated. The a... more Certain affine sets arising from spreads of the projective space PG(3, q) are investigated. The affine set arising from a Lüneburg spread is studied in detail.
We describe a simple method to construct many different linear codes arising from caps in a proje... more We describe a simple method to construct many different linear codes arising from caps in a projective space whose automorphism group is known in advance. This procedure begins with a fairly small point set in an appropriate finite projective space.
The chords of a twisted cubic in PG(3,q) are mapped via their Plücker coordinates to the points o... more The chords of a twisted cubic in PG(3,q) are mapped via their Plücker coordinates to the points of a Veronese surface lying on the Klein quadric in PG(5,q). This correspondence over a finite field gives a cap in PG(5,q), that is, a set of points no three of which are collinear. The dual structure, namely the axes of the osculating developable, is also mapped to a Veronese surface. The two surfaces can be combined to give a larger cap. The constructions can be extended to the chords and axes of an arbitrary (q+1)-arc in PG(3,q) when q is even. An alternative construction for the cap associated to a twisted cubic is given for q odd. [For part I and II see A. A. Bruen and J. W. P. Hirschfeld, Geom. Dedicata 6, 495-509 (1977; Zbl 0394.51005) and Geom. Dedicata 7, 333-353 (1978; Zbl 0394.51006)].
The genus of a maximal curve over a finite field with r^2 elements is either g_0=r(r-1)/2 or less... more The genus of a maximal curve over a finite field with r^2 elements is either g_0=r(r-1)/2 or less than or equal to g_1=(r-1)^2/4. Maximal curves with genus g_0 or g_1 have been characterized up to isomorphism. A natural genus to be studied is g_2=(r-1)(r-3)/8, and for this genus there are two non-isomorphism maximal curves known when r \equiv 3 (mod
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Papers by Antonio Cossidente