36 research outputs found
Unitals in with a large 2-point stabiliser
Let \cU be a unital embedded in the Desarguesian projective plane
\PG(2,q^2). Write for the subgroup of \PGL(3,q^2) which preserves
\cU. We show that \cU is classical if and only if \cU has two distinct
points for which the stabiliser has order .Comment: Revised version - clarified the case mu\neq\lambda^{q+1} - 7 page
Construction of a 3-Dimensional MDS code
In this paper, we describe a procedure for constructing --ary
--MDS codes, of length (for odd) or (for
even), using a set of non--degenerate Hermitian forms in .Comment: 8 Pages Minor typesetting changes; dedicatio
Grassmannians of codes
Consider the point line-geometry having as points all
the -linear codes having minimum dual distance at least and where
two points and are collinear whenever is a -linear
code having minimum dual distance at least . We are interested in the
collinearity graph of The graph
is a subgraph of the Grassmann graph and also a subgraph of
the graph of the linear codes having minimum dual distance at
least introduced in~[M. Kwiatkowski, M. Pankov, On the distance between
linear codes, Finite Fields Appl. 39 (2016), 251--263,
doi:10.1016/j.ffa.2016.02.004, arXiv:1506.00215]. We shall study the structure
of in relation to that of and we will
characterize the set of its isolated vertices. We will then focus on
and providing necessary and sufficient
conditions for them to be connected.Comment: 20 pages/minor corrections/updated bibliograph
Geometries arising from trilinear forms on low-dimensional vector spaces
Let Gk(V) be the k-Grassmannian of a vector space V with dimV=n. Given a hyperplane H of Gk(V), we define in [I. Cardinali, L. Giuzzi, A. Pasini, A geometric approach to alternating k-linear forms, J. Algebraic Combin. doi: 10.1007/s10801-016-0730-6] a point-line subgeometry of PG(V) called the geometry of poles of H. In the present paper, exploiting the classification of alternating trilinear forms in low dimension, we characterize the possible geometries of poles arising for k=3 and n≤7 and propose some new constructions. We also extend a result of [J.Draisma, R. Shaw, Singular lines of trilinear forms, Linear Algebra Appl. doi: 10.1016/j.laa.2010.03.040] regarding the existence of line spreads of PG(5,K) arising from hyperplanes of G3(V)
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 . 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).Comment: 17 page
New results on path-decompositions and their down-links
In a recent paper the concept of \emph{down-link} from a
-design \cB to a -design \cB' has been
introduced. In the present paper the spectrum problems for
are studied. General results on the existence of path-decompositions
and embeddings between path-decompositions playing a fundamental role for the construction of down-links
are also presented
On the generation of some Lie-type geometries
Let be a building of Coxeter type or or defined over a given division ring (a field when ). For a non-connected set of nodes of the diagram , let be the -grassmannian of . We prove that cannot be generated over any proper sub-division ring of . As a consequence, the generating rank of is infinite when is not finitely generated. In particular, if is the algebraic closure of a finite field of prime order then the generating rank of is infinite, although its embedding rank is either or
On the generation of some Lie-type geometries
Let Xn(K) be a building of Coxeter type Xn=An or Xn=Dn defined over a given division ring K (a field when Xn=Dn). For a non-connected set J of nodes of the diagram Xn, let Γ(K)=GrJ(Xn(K)) be the J-grassmannian of Xn(K). We prove that Γ(K) cannot be generated over any proper sub-division ring K0 of K. As a consequence, the generating rank of Γ(K) is infinite when K is not finitely generated. In particular, if K is the algebraic closure of a finite field of prime order then the generating rank of Gr1,n(An(K)) is infinite, although its embedding rank is either (n+1)2−1 or (n+1)2