arXiv subjects

A. Aguglia

Publications and source records attributed to A. Aguglia.

7 recordsLinked to original sources

arXiv2017

A characterization of Hermitian varieties as codewords

It is known that the Hermitian varieties are codewords in the code defined by the points and hyperplanes of the projective spaces $PG(r,q^2)$. In finite geometry, also quasi-Hermitian varieties are defined. These are sets of points of $PG(r,q^2)$ of the same size as a non-singular Hermitian variety of $PG(r,q^2)$, having the same intersection sizes with the hyperplanes of $PG(r,q^2)$. In the planar case, this reduces to the definition of a unital. A famous result of Blokhuis, Brouwer, and Wilbrink states that every unital in the code of the points and lines of $PG(2,q^2)$ is a Hermitian curve. We prove a similar result for the quasi-Hermitian varieties in $PG(3,q^2)$, $q=p^{h}$, as well as in $PG(r,q^2)$, $q=p$ prime, or $q=p^2$, $p$ prime, and $r\geq 4$.

math.CO↗

arXiv2009

An alternative construction of B-M and B-T unitals in Desarguesian planes

We present a new construction of non-classical unitals from a classical unital $U$ in $PG(2,q^2)$. The resulting non-classical unitals are B-M unitals. The idea is to find a non-standard model $Π$ of $PG(2,q^2)$ with the following three properties: 1. points of $Π$ are those of $PG(2,q^2)$; 2. lines of $Π$ are certain lines and conics of $PG(2,q^2)$; 3. the points in $U$ form a non-classical B-M unital in $Π$. Our construction also works for the B-T unital, provided that conics are replaced by certain algebraic curves of higher degree.

math.AG↗

arXiv2008

On the non--existence of certain hyperovals in dual André planes of order $2^{2h}$

No regular hyperoval of the Desarguesian affine plane $AG(2,2^{2h})$, with $h>1$, is inherited by a dual André plane of order $2^{2h}$ with dimension 2 over its centre.

math.CO↗

arXiv2007

Construction of a 3-Dimensional MDS code

In this paper, we describe a procedure for constructing $q$--ary $[N,3,N-2]$--MDS codes, of length $N\leq q+1$ (for $q$ odd) or $N\leq q+2$ (for $q$ even), using a set of non--degenerate Hermitian forms in $PG(2,q^2)$.

cs.IT↗

arXiv2007

Algebraic curves and Maximal arcs

A lower bound on the minimum degree of the plane algebraic curves containing every point in a large point-set $K$ of the Desarguesian plane $PG(2,q)$ is obtained. The case where $K$ is a maximal $(k,n)$-arc is considered to greater extent.

math.CO↗

arXiv2007

Orthogonal arrays from Hermitian varieties

An orthogonal array OA(q^{2n-1},q^{2n-2}, q,2) is constructed from the action of a subset of PGL(n+1,q^2) on some non--degenerate Hermitian varieties in PG(n,q^2). It is also shown that the rows of this orthogonal array correspond to some blocks of an affine design, which for q> 2 is a non--classical model of the affine space AG(2n-1,q).

math.CO↗

arXiv2007

An algorithm for constructing some maximal arcs in $\PG(2,q^2)$

In 1974, J. Thas constructed a new class of maximal arcs for the Desarguesian plane of order $q^2$. The construction relied upon the existence of a regular spread of tangent lines to an ovoid in $\PG(3,q)$ and, in particular, it does apply to the Suzuki--Tits ovoid. In this paper, we describe an algorithm for obtaining a possible representation of such arcs in $\PG(2,q^2)$.

math.CO↗