The sets of the absolute points of (possibly degenerate) polarities of a projective space are wel... more The sets of the absolute points of (possibly degenerate) polarities of a projective space are well known. The sets of the absolute points of (possibly degenerate) correlations, different from polarities, of $$\mathrm {PG}(2,q^n)$$ PG ( 2 , q n ) , have been completely determined by B.C. Kestenband in 11 papers from 1990 to 2014, for non-degenerate correlations and by D’haeseleer and Durante (Electron J Combin 27(2):2–32, 2020) for degenerate correlations. The sets of the absolute points of degenerate correlations, different from degenerate polarities, of a projective space $$\mathrm {PG}(3,q^n)$$ PG ( 3 , q n ) have been classified in (Donati and Durante in J Algebr Comb 54:109–133, 2021). In this paper, we consider the four dimensional case and completely determine the sets of the absolute points of degenerate correlations, different from degenerate polarities, of a projective space $$\mathrm {PG}(4,q^n).$$ PG ( 4 , q n ) . As an application, we show that some of these sets are rel...
Dedicated to Adriano Barlotti on the occasion of his 80th birthday Abstract. The projective full ... more Dedicated to Adriano Barlotti on the occasion of his 80th birthday Abstract. The projective full embeddings of partial geometries are known. So are the projective full embeddings of semipartial and dual semipartial geometries in case of a> 1. If a 1, a semipartial geometry is known as a partial quadrangle. No projective full embedding of a proper partial quadrangle is known. However besides a unique example for q 2, there is one example known of a dual partial quadrangle fully embedded in a PGð3; qÞ, any q. In this paper we will prove that if the dual of a proper partial quadrangle S is fully embedded in PGð3; qÞ, then m c q. If equality holds, then S is uniquely defined. q tþ1 1
Let $V$ be a $(d+1)$-dimensional vector space over a field $\mathbb{F}$. Sesquilinear forms over ... more Let $V$ be a $(d+1)$-dimensional vector space over a field $\mathbb{F}$. Sesquilinear forms over $V$ have been largely studied when they are reflexive and hence give rise to a (possibly degenerate) polarity of the $d$-dimensional projective space PG$(V)$. Everything is known in this case for both degenerate and non-degenerate reflexive forms if $\mathbb{F}$ is either ${\mathbb{R}}$, ${\mathbb{C}}$ or a finite field ${\mathbb{F}}_q$. In this paper we consider degenerate, non-reflexive sesquilinear forms of $V=\mathbb{F}_{q^n}^3$. We will see that these forms give rise to degenerate correlations of PG$(2,q^n)$ whose set of absolute points are, besides cones, the (possibly degenerate) $C_F^m$-sets. In the final section we collect some results from the huge work of B.C. Kestenband regarding what is known for the set of the absolute points of correlations in PG$(2,q^n)$ induced by a non-degenerate, non-reflexive sesquilinear form of $V=\mathbb{F}_{q^n}^3$.
In this paper we study sets X of points of both ane and projective spaces over the Galois eld GF(... more In this paper we study sets X of points of both ane and projective spaces over the Galois eld GF( q) such that every line of the geometry that is neither contained in X nor disjoint from X meets the set X in a constant number of points and we determine all such sets. This study has its main motivation in connection with a recent study of neighbour transitive codes in Johnson graphs by Liebler and Praeger [Designs, Codes and Crypt., 2014]. We prove that, up to complements, in PG(n;q) such a set X is either a subspace or n = 2;q is even and X is a maximal arc of degree m. In AG(n;q) we show that X is either the union of parallel hyperplanes or a cylinder with base a maximal arc of degree m (or the complement of a maximal arc) or a cylinder with base the projection of a quadric. Finally we show that in the ane case there are examples (dierent from subspaces or their complements) in AG(n; 4) and in AG(n; 16) giving new neighbour transitive codes in Johnson graphs.
Let K be the Galois field Fqt of order q , q = p, p a prime, A = Aut(K) be the automorphism group... more Let K be the Galois field Fqt of order q , q = p, p a prime, A = Aut(K) be the automorphism group of K and σ = (σ0, . . . , σd−1) ∈ A , d ≥ 1. In this paper the following generalization of the Veronese map is studied: νd,σ : 〈v〉 ∈ PG(n− 1,K) −→ 〈v σ0 ⊗ v1 ⊗ · · · ⊗ vd−1 〉 ∈ PG(n − 1,K). Its image will be called the (d,σ)-Veronese variety Vd,σ. For d = t, σ a generator of Gal(Fqt |Fq) and σ = (1, σ, σ , . . . , σ), the (t,σ)-Veronese variety Vt,σ is the variety studied in [19, 12, 14] and it will be denoted by Vt,σ. Such a variety is the Grassmann embedding of the Desarguesian spread of PG(nt−1, Fq) and it has been used to construct codes [6] and (partial) ovoids of quadrics, see [12, 15]. We will show that Vd,σ is the Grassmann embedding of a normal rational scroll and we will prove that it has the property that any d + 1 points of it are linearly independent. As applications we give a characterization of d + 2 linearly dependent points of Vd,σ and we show how such a property is int...
The prqjective full embeddings of partial geometries are known. So are the projective full embedd... more The prqjective full embeddings of partial geometries are known. So are the projective full embeddings of semipartial and dual semipartial geometries in case of α > 1. If α = 1, a semipartial geometry is known as a partial quadrangle. No projective full embedding of a proper partial quadrangle is known. However besides a unique example for q = 2, there is one example known of a dual partial quadrangle fully embedded in a PG(3, q), any q. In this paper we will prove that if the dual of a proper partial quadrangle £f is fully embedded in PG(3, q), then μ ^ q -^ . If equality holds, then 5^ is uniquely defined.
Algebraic pencils of surfaces in a three–dimensional circle geometry are used to construct severa... more Algebraic pencils of surfaces in a three–dimensional circle geometry are used to construct several infinite families of non-André subregular translation planes which are three–dimensional over their kernels. In fact, exponentially many such planes of a given order are constructed for both even and odd characteristic.
Let A and B be two points of $$\mathop {\mathrm{PG}}(d,q^n)$$PG(d,qn) and let $$\Phi $$Φ be a col... more Let A and B be two points of $$\mathop {\mathrm{PG}}(d,q^n)$$PG(d,qn) and let $$\Phi $$Φ be a collineation between the stars of lines with vertices A and B, that does not map the line AB into itself. In this paper we prove that if $$d=2$$d=2 or $$d\ge 3$$d≥3 and the lines $$\Phi ^{-1}(AB), AB, \Phi (AB) $$Φ-1(AB),AB,Φ(AB) are not in a common plane, then the set $$\mathcal{C}$$C of points of intersection of corresponding lines under $$\Phi $$Φ is the union of $$q-1$$q-1 scattered $${\mathbb {F}}_{q}$$Fq-linear sets of rank n together with $$\{A,B\}$${A,B}. As an application we will construct, starting from the set $$\mathcal{C}$$C, infinite families of non-linear $$(d+1, n, q;d-1)$$(d+1,n,q;d-1)-MRD codes, $$d\le n-1$$d≤n-1, generalizing those recently constructed in Cossidente et al. (Des Codes Cryptogr 79:597–609, 2016) and Durante and Siciliano (Electron J Comb, 2017).
In this paper a description for sets in $${\mathrm {PG}}(3, q)$$PG(3,q) of type (q, n) with respe... more In this paper a description for sets in $${\mathrm {PG}}(3, q)$$PG(3,q) of type (q, n) with respect to planes is given.
The sets of the absolute points of (possibly degenerate) polarities of a projective space are wel... more The sets of the absolute points of (possibly degenerate) polarities of a projective space are well known. The sets of the absolute points of (possibly degenerate) correlations, different from polarities, of $${{\mathrm{PG}}}(2,q^n)$$ PG ( 2 , q n ) , have been completely determined by B.C. Kestenband in 11 papers from 1990 to 2014, for non-degenerate correlations and by D’haeseleer and Durante (Electron J Combin 27(2):2–32, 2020) for degenerate correlations. In this paper, we completely determine the sets of the absolute points of degenerate correlations, different from degenerate polarities, of a projective space $${{\mathrm{PG}}}(3,q^n)$$ PG ( 3 , q n ) . As an application we show that, for q even, some of these sets are related to the Segre’s $$(2^h+1)$$ ( 2 h + 1 ) -arc of $${{\mathrm{PG}}}(3,2^n)$$ PG ( 3 , 2 n ) and to the Lüneburg spread of $${{\mathrm{PG}}}(3,2^{2h+1})$$ PG ( 3 , 2 2 h + 1 ) .
Let $V$ be a $(d+1)$-dimensional vector space over a field $\mathbb{F}$. Sesquilinear forms over... more Let $V$ be a $(d+1)$-dimensional vector space over a field $\mathbb{F}$. Sesquilinear forms over $V$ have been largely studied when they are reflexive and hence give rise to a (possibly degenerate) polarity of the $d$-dimensional projective space $\mathrm{PG}(V)$. Everything is known in this case for both degenerate and non-degenerate reflexive forms if $\mathbb{F}$ is either ${\mathbb R}$, ${\mathbb C}$ or a finite field ${\mathbb F}_q$. In this paper we consider degenerate, non-reflexive sesquilinear forms of $V=\mathbb{F}_{q^n}^3$. We will see that these forms give rise to degenerate correlations of $\mathrm{PG}(2,q^n)$ whose set of absolute points are, besides cones, the (possibly degenerate) $C_F^m$-sets studied by Donati and Durante in 2014. In the final section we collect some results from the huge work of B.C. Kestenband regarding what is known for the set of the absolute points of correlations in $\mathrm{PG}(2,q^n)$ induced by a non-degenerate, non-reflex...
In this paper we construct infinite families of non-linear maximum rank distance codes by using t... more In this paper we construct infinite families of non-linear maximum rank distance codes by using the setting of bilinear forms of a finite vector space. We also give a geometric description of such codes by using the cyclic model for the field reduction of finite geometries and we show that these families contain the non-linear maximum rank distance codes recently provided by Cossidente, Marino and Pavese.
The sets of the absolute points of (possibly degenerate) polarities of a projective space are wel... more The sets of the absolute points of (possibly degenerate) polarities of a projective space are well known. The sets of the absolute points of (possibly degenerate) correlations, different from polarities, of $$\mathrm {PG}(2,q^n)$$ PG ( 2 , q n ) , have been completely determined by B.C. Kestenband in 11 papers from 1990 to 2014, for non-degenerate correlations and by D’haeseleer and Durante (Electron J Combin 27(2):2–32, 2020) for degenerate correlations. The sets of the absolute points of degenerate correlations, different from degenerate polarities, of a projective space $$\mathrm {PG}(3,q^n)$$ PG ( 3 , q n ) have been classified in (Donati and Durante in J Algebr Comb 54:109–133, 2021). In this paper, we consider the four dimensional case and completely determine the sets of the absolute points of degenerate correlations, different from degenerate polarities, of a projective space $$\mathrm {PG}(4,q^n).$$ PG ( 4 , q n ) . As an application, we show that some of these sets are rel...
Dedicated to Adriano Barlotti on the occasion of his 80th birthday Abstract. The projective full ... more Dedicated to Adriano Barlotti on the occasion of his 80th birthday Abstract. The projective full embeddings of partial geometries are known. So are the projective full embeddings of semipartial and dual semipartial geometries in case of a> 1. If a 1, a semipartial geometry is known as a partial quadrangle. No projective full embedding of a proper partial quadrangle is known. However besides a unique example for q 2, there is one example known of a dual partial quadrangle fully embedded in a PGð3; qÞ, any q. In this paper we will prove that if the dual of a proper partial quadrangle S is fully embedded in PGð3; qÞ, then m c q. If equality holds, then S is uniquely defined. q tþ1 1
Let $V$ be a $(d+1)$-dimensional vector space over a field $\mathbb{F}$. Sesquilinear forms over ... more Let $V$ be a $(d+1)$-dimensional vector space over a field $\mathbb{F}$. Sesquilinear forms over $V$ have been largely studied when they are reflexive and hence give rise to a (possibly degenerate) polarity of the $d$-dimensional projective space PG$(V)$. Everything is known in this case for both degenerate and non-degenerate reflexive forms if $\mathbb{F}$ is either ${\mathbb{R}}$, ${\mathbb{C}}$ or a finite field ${\mathbb{F}}_q$. In this paper we consider degenerate, non-reflexive sesquilinear forms of $V=\mathbb{F}_{q^n}^3$. We will see that these forms give rise to degenerate correlations of PG$(2,q^n)$ whose set of absolute points are, besides cones, the (possibly degenerate) $C_F^m$-sets. In the final section we collect some results from the huge work of B.C. Kestenband regarding what is known for the set of the absolute points of correlations in PG$(2,q^n)$ induced by a non-degenerate, non-reflexive sesquilinear form of $V=\mathbb{F}_{q^n}^3$.
In this paper we study sets X of points of both ane and projective spaces over the Galois eld GF(... more In this paper we study sets X of points of both ane and projective spaces over the Galois eld GF( q) such that every line of the geometry that is neither contained in X nor disjoint from X meets the set X in a constant number of points and we determine all such sets. This study has its main motivation in connection with a recent study of neighbour transitive codes in Johnson graphs by Liebler and Praeger [Designs, Codes and Crypt., 2014]. We prove that, up to complements, in PG(n;q) such a set X is either a subspace or n = 2;q is even and X is a maximal arc of degree m. In AG(n;q) we show that X is either the union of parallel hyperplanes or a cylinder with base a maximal arc of degree m (or the complement of a maximal arc) or a cylinder with base the projection of a quadric. Finally we show that in the ane case there are examples (dierent from subspaces or their complements) in AG(n; 4) and in AG(n; 16) giving new neighbour transitive codes in Johnson graphs.
Let K be the Galois field Fqt of order q , q = p, p a prime, A = Aut(K) be the automorphism group... more Let K be the Galois field Fqt of order q , q = p, p a prime, A = Aut(K) be the automorphism group of K and σ = (σ0, . . . , σd−1) ∈ A , d ≥ 1. In this paper the following generalization of the Veronese map is studied: νd,σ : 〈v〉 ∈ PG(n− 1,K) −→ 〈v σ0 ⊗ v1 ⊗ · · · ⊗ vd−1 〉 ∈ PG(n − 1,K). Its image will be called the (d,σ)-Veronese variety Vd,σ. For d = t, σ a generator of Gal(Fqt |Fq) and σ = (1, σ, σ , . . . , σ), the (t,σ)-Veronese variety Vt,σ is the variety studied in [19, 12, 14] and it will be denoted by Vt,σ. Such a variety is the Grassmann embedding of the Desarguesian spread of PG(nt−1, Fq) and it has been used to construct codes [6] and (partial) ovoids of quadrics, see [12, 15]. We will show that Vd,σ is the Grassmann embedding of a normal rational scroll and we will prove that it has the property that any d + 1 points of it are linearly independent. As applications we give a characterization of d + 2 linearly dependent points of Vd,σ and we show how such a property is int...
The prqjective full embeddings of partial geometries are known. So are the projective full embedd... more The prqjective full embeddings of partial geometries are known. So are the projective full embeddings of semipartial and dual semipartial geometries in case of α > 1. If α = 1, a semipartial geometry is known as a partial quadrangle. No projective full embedding of a proper partial quadrangle is known. However besides a unique example for q = 2, there is one example known of a dual partial quadrangle fully embedded in a PG(3, q), any q. In this paper we will prove that if the dual of a proper partial quadrangle £f is fully embedded in PG(3, q), then μ ^ q -^ . If equality holds, then 5^ is uniquely defined.
Algebraic pencils of surfaces in a three–dimensional circle geometry are used to construct severa... more Algebraic pencils of surfaces in a three–dimensional circle geometry are used to construct several infinite families of non-André subregular translation planes which are three–dimensional over their kernels. In fact, exponentially many such planes of a given order are constructed for both even and odd characteristic.
Let A and B be two points of $$\mathop {\mathrm{PG}}(d,q^n)$$PG(d,qn) and let $$\Phi $$Φ be a col... more Let A and B be two points of $$\mathop {\mathrm{PG}}(d,q^n)$$PG(d,qn) and let $$\Phi $$Φ be a collineation between the stars of lines with vertices A and B, that does not map the line AB into itself. In this paper we prove that if $$d=2$$d=2 or $$d\ge 3$$d≥3 and the lines $$\Phi ^{-1}(AB), AB, \Phi (AB) $$Φ-1(AB),AB,Φ(AB) are not in a common plane, then the set $$\mathcal{C}$$C of points of intersection of corresponding lines under $$\Phi $$Φ is the union of $$q-1$$q-1 scattered $${\mathbb {F}}_{q}$$Fq-linear sets of rank n together with $$\{A,B\}$${A,B}. As an application we will construct, starting from the set $$\mathcal{C}$$C, infinite families of non-linear $$(d+1, n, q;d-1)$$(d+1,n,q;d-1)-MRD codes, $$d\le n-1$$d≤n-1, generalizing those recently constructed in Cossidente et al. (Des Codes Cryptogr 79:597–609, 2016) and Durante and Siciliano (Electron J Comb, 2017).
In this paper a description for sets in $${\mathrm {PG}}(3, q)$$PG(3,q) of type (q, n) with respe... more In this paper a description for sets in $${\mathrm {PG}}(3, q)$$PG(3,q) of type (q, n) with respect to planes is given.
The sets of the absolute points of (possibly degenerate) polarities of a projective space are wel... more The sets of the absolute points of (possibly degenerate) polarities of a projective space are well known. The sets of the absolute points of (possibly degenerate) correlations, different from polarities, of $${{\mathrm{PG}}}(2,q^n)$$ PG ( 2 , q n ) , have been completely determined by B.C. Kestenband in 11 papers from 1990 to 2014, for non-degenerate correlations and by D’haeseleer and Durante (Electron J Combin 27(2):2–32, 2020) for degenerate correlations. In this paper, we completely determine the sets of the absolute points of degenerate correlations, different from degenerate polarities, of a projective space $${{\mathrm{PG}}}(3,q^n)$$ PG ( 3 , q n ) . As an application we show that, for q even, some of these sets are related to the Segre’s $$(2^h+1)$$ ( 2 h + 1 ) -arc of $${{\mathrm{PG}}}(3,2^n)$$ PG ( 3 , 2 n ) and to the Lüneburg spread of $${{\mathrm{PG}}}(3,2^{2h+1})$$ PG ( 3 , 2 2 h + 1 ) .
Let $V$ be a $(d+1)$-dimensional vector space over a field $\mathbb{F}$. Sesquilinear forms over... more Let $V$ be a $(d+1)$-dimensional vector space over a field $\mathbb{F}$. Sesquilinear forms over $V$ have been largely studied when they are reflexive and hence give rise to a (possibly degenerate) polarity of the $d$-dimensional projective space $\mathrm{PG}(V)$. Everything is known in this case for both degenerate and non-degenerate reflexive forms if $\mathbb{F}$ is either ${\mathbb R}$, ${\mathbb C}$ or a finite field ${\mathbb F}_q$. In this paper we consider degenerate, non-reflexive sesquilinear forms of $V=\mathbb{F}_{q^n}^3$. We will see that these forms give rise to degenerate correlations of $\mathrm{PG}(2,q^n)$ whose set of absolute points are, besides cones, the (possibly degenerate) $C_F^m$-sets studied by Donati and Durante in 2014. In the final section we collect some results from the huge work of B.C. Kestenband regarding what is known for the set of the absolute points of correlations in $\mathrm{PG}(2,q^n)$ induced by a non-degenerate, non-reflex...
In this paper we construct infinite families of non-linear maximum rank distance codes by using t... more In this paper we construct infinite families of non-linear maximum rank distance codes by using the setting of bilinear forms of a finite vector space. We also give a geometric description of such codes by using the cyclic model for the field reduction of finite geometries and we show that these families contain the non-linear maximum rank distance codes recently provided by Cossidente, Marino and Pavese.
Uploads
Papers by Nicola Durante