Stability of Erd\H{o}s-Ko-Rado Theorems in Circle Geometries

Circle geometries are incidence structures that capture the geometry of circles on spheres, cones and hyperboloids in 3-dimensional space. In a previous paper, the author characterised the largest intersecting families in finite ovoidal circle geometries, except for M\"obius planes of odd order. In this paper we show that also in these M\"obius planes, if the order is greater than 3, the largest intersecting families are the sets of circles through a fixed point. We show the same result in the only known family of finite non-ovoidal circle geometries. Using the same techniques, we show a stability result on large intersecting families in all ovoidal circle geometries. More specifically, we prove that an intersecting family $\mathcal F$ in one of the known finite circle geometries of order $q$, with $|\mathcal F| \geq \frac 1 {\sqrt2} q^2 + 2 \sqrt 2 q + 8$, must consist of circles through a common point, or through a common nucleus in case of a Laguerre plane of even order.Comment: 18 page

Small weight codewords of projective geometric codes II

The $p$-ary linear code $\mathcal C_{k}(n,q)$ is defined as the row space of the incidence matrix $A$ of $k$-spaces and points of $\text{PG}(n,q)$. It is known that if $q$ is square, a codeword of weight $q^k\sqrt{q}+\mathcal O \left( q^{k-1} \right)$ exists that cannot be written as a linear combination of at most $\sqrt{q}$ rows of $A$. Over the past few decades, researchers have put a lot of effort towards proving that any codeword of smaller weight does meet this property. We show that if $q \geqslant 32$ is a composite prime power, every codeword of $\mathcal C_k(n,q)$ up to weight $\mathcal O \left( {q^k\sqrt{q}} \right)$ is a linear combination of at most $\sqrt{q}$ rows of $A$. We also generalise this result to the codes $\mathcal C_{j,k}(n,q)$, which are defined as the $p$-ary row span of the incidence matrix of $k$-spaces and $j$-spaces, $j < k$.Comment: 22 page

On additive MDS codes with linear projections

We support some evidence that a long additive MDS code over a finite field must be equivalent to a linear code. More precisely, let $C$ be an $\mathbb F_q$-linear $(n,q^{hk},n-k+1)_{q^h}$ MDS code over $\mathbb F_{q^h}$. If $k=3$, $h \in \{2,3\}$, $n > \max \{q^{h-1},h q -1\} + 3$, and $C$ has three coordinates from which its projections are equivalent to linear codes, we prove that $C$ itself is equivalent to a linear code. If $k>3$, $n > q+k$, and there are two disjoint subsets of coordinates whose combined size is at most $k-2$ from which the projections of $C$ are equivalent to linear codes, we prove that $C$ is equivalent to a code which is linear over a larger field than $\mathbb F_q$.Comment: 15 page

On the minimum size of linear sets

Recently, a lower bound was established on the size of linear sets in projective spaces, that intersect a hyperplane in a canonical subgeometry. There are several constructions showing that this bound is tight. In this paper, we generalize this bound to linear sets meeting some subspace $\pi$ in a canonical subgeometry. We obtain a tight lower bound on the size of any $\mathbb F_q$-linear set spanning $\text{PG}(d,q^n)$ in case that $n \leq q$ and $n$ is prime. We also give constructions of linear sets attaining equality in the former bound, both in the case that $\pi$ is a hyperplane, and in the case that $\pi$ is a lower dimensional subspace.Comment: 24 pages; the updated version contains a consequences of the recent paper by Csajb\'ok, Marino and Pepe (arXiv:2306.07488), providing a tight lower bound in some case

Association schemes and orthogonality graphs on anisotropic points of polar spaces

In this paper, we study association schemes on the anisotropic points of classical polar spaces. Our main result concerns non-degenerate elliptic and hyperbolic quadrics in PG$(n,q)$ with $q$ odd. We define relations on the anisotropic points of such a quadric that depend on the type of line spanned by the points and whether or not they are of the same "quadratic type". This yields an imprimitive $5$-class association scheme. We calculate the matrices of eigenvalues and dual eigenvalues of this scheme. We also use this result, together with similar results from the literature concerning other classical polar spaces, to exactly calculate the spectrum of orthogonality graphs on the anisotropic points of non-degenerate quadrics in odd characteristic and of non-degenerate Hermitian varieties. As a byproduct, we obtain a $3$-class association scheme on the anisotropic points of non-degenerate Hermitian varieties, where the relation containing two points depends on the type of line spanned by these points, and whether or not they are orthogonal.Comment: 29 pages. Update: an extra reference was adde

Additive MDS codes

We prove that an additive code over a finite field which has a few projections which are equivalent to a linear code is itself equivalent to a linear code, providing the code is not too short.Postprint (published version

The minimum weight of the code of intersecting lines in PG(3,q){\rm PG}(3,q)

We characterise the minimum weight codewords of the $p$-ary linear code of intersecting lines in ${\rm PG}(3,q)$, $q=p^h$, $q\geq19$, $p$ prime, $h\geq 1$. If $q$ is even, the minimum weight equals $q^3+q^2+q+1$. If $q$ is odd, the minimum weight equals $q^3+2q^2+q+1$. For $q$ even, we also characterise the codewords of second smallest weight.Comment: 12 page

Blocking subspaces with points and hyperplanes

In this paper, we characterise the smallest sets $B$ consisting of points and hyperplanes in $\text{PG}(n,q)$, such that each $k$-space is incident with at least one element of $B$. If $k > \frac {n-1} 2$, then the smallest construction consists only of points. Dually, if $k < \frac{n-1}2$, the smallest example consists only of hyperplanes. However, if $k = \frac{n-1}2$, then there exist sets containing both points and hyperplanes, which are smaller than any blocking set containing only points or only hyperplanes.Comment: 7 pages. UPDATE: After publication of this paper, we found out that in case $k = \frac{n-1}2$, the correct lower bound and a classification of the smallest examples was already established by Blokhuis, Brouwer, and Sz\H{o}nyi [A. Blokhuis, A. E. Brouwer, T. Sz\H{o}nyi. On the chromatic number of $q$-Kneser graphs. Des. Codes Crytpogr. 65:187-197, 2012

On mm-ovoids of Q+(7,q)Q^+(7,q) with qq odd

In this paper, we provide a construction of $(q+1)$-ovoids of the hyperbolic quadric $Q^+(7,q)$, $q$ an odd prime power, by glueing $(q+1)/2$-ovoids of the elliptic quadric $Q^-(5,q)$. This is possible by controlling some intersection properties of (putative) $m$-ovoids of elliptic quadrics. It yields eventually $(q+1)$-ovoids of $Q^+(7,q)$ not coming from a $1$-system. Secondly, we also construct $m$-ovoids for $m \in \{ 2,4,6,8,10\}$ in $Q^+(7,3)$. Therefore we first investigate how to construct spreads of \pg(3,q) that have as many secants to an elliptic quadric as possible

Small weight code words arising from the incidence of points and hyperplanes in PG(n,q\boldsymbol{n,q})

Let $C_{n-1}(n,q)$ be the code arising from the incidence of points and hyperplanes in the Desarguesian projective space PG($n,q$). Recently, Polverino and Zullo proved that within this code, all non-zero code words of weight at most $2q^{n-1}$ are scalar multiples of either the incidence vector of one hyperplane, or the difference of the incidence vectors of two distinct hyperplanes. We improve this result, proving that when $q>17$ and $q\notin\{25,27,29,31,32,49,121\}$, all code words of weight at most $(4q-\sqrt{8q}-\frac{33}{2})q^{n-2}$ are linear combinations of incidence vectors of hyperplanes through a fixed $(n-3)$-space. Depending on the omitted value for $q$, we can lower the bound on the weight of $c$ to obtain the same results