Unitals in PG(2,q2)PG(2,q^2) with a large 2-point stabiliser

Let \cU be a unital embedded in the Desarguesian projective plane \PG(2,q^2). Write $M$ 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 $P,Q$ for which the stabiliser $G=M_{P,Q}$ has order $q^2-1$.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 $q$--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 $PG(2,q^2)$.Comment: 8 Pages Minor typesetting changes; dedicatio

Grassmannians of codes

Consider the point line-geometry ${\mathcal P}_t(n,k)$ having as points all the $[n,k]$-linear codes having minimum dual distance at least $t+1$ and where two points $X$ and $Y$ are collinear whenever $X\cap Y$ is a $[n,k-1]$-linear code having minimum dual distance at least $t+1$. We are interested in the collinearity graph $\Lambda_t(n,k)$ of ${\mathcal P}_t(n,k).$ The graph $\Lambda_t(n,k)$ is a subgraph of the Grassmann graph and also a subgraph of the graph $\Delta_t(n,k)$ of the linear codes having minimum dual distance at least $t+1$ 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 $\Lambda_t(n,k)$ in relation to that of $\Delta_t(n,k)$ and we will characterize the set of its isolated vertices. We will then focus on $\Lambda_1(n,k)$ and $\Lambda_2(n,k)$ 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 $q^2$. 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).Comment: 17 page

New results on path-decompositions and their down-links

In a recent paper the concept of \emph{down-link} from a $(K_v,\Gamma)$-design \cB to a $(K_n,\Gamma')$-design \cB' has been introduced. In the present paper the spectrum problems for $\Gamma'=P_4$ 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 $X_n(K)$ be a building of Coxeter type $X_n=A_n$ or $X_n=D_n$ or defined over a given division ring $K$ (a field when $X_n=D_n$). For a non-connected set $J$ of nodes of the diagram $X_n$, let $\Gamma(K)=Gr_J(X_n(K))$ be the $J$-grassmannian of $X_n(K)$ . We prove that $\Gamma(K)$ cannot be generated over any proper sub-division ring $K_0$ of $K$ . As a consequence, the generating rank of $\Gamma(K)$ 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 $Gr_{1,n}(A_n(K))$ is infinite, although its embedding rank is either $(n+1)^2-1$ or $(n+1)^2$

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