Abstract We prove new upper bounds on the smallest size of affine blocking sets, that is, sets points in a finite space intersect every subspace fixed codimension. show an equivalence between with respect to codimension‐2 subspaces are generated by taking union lines through origin, and strong corresponding projective space, which turn equivalent minimal codes. Using this equivalence, we improve current best set spaces over fields at least 3. Furthermore, using coding theoretic techniques,...

A strong blocking set in a finite projective space is of points that intersects each hyperplane spanning set. We provide new graph theoretic construction such sets: combining constant-degree expanders with asymptotically good codes, we explicitly construct sets the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis k minus 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo>...

We study hyperplane covering problems for finite grid-like structures in $\mathbb{R}^d$. call a set $\mathcal{C}$ of points $\mathbb{R}^2$ conical grid if the line $y = a_i$ intersects exactly $i$ points, some $a_1 > \cdots a_n \in \mathbb{R}$. prove that number lines required to cover every point such at least $k$ times is $nk\left(1-\frac{1}{e}-O(\frac{1}{n}) \right)$. If obtained by cutting an $m \times n$ into half along one diagonals, then we lower bound...

A 1993 result of Alon and Füredi gives a sharp upper bound on the number zeros multivariate polynomial over an integral domain in finite grid, terms degree polynomial. This was recently generalized to polynomials arbitrary commutative ring, assuming certain ‘Condition (D)’ grid which holds vacuously when ring is domain. In first half this paper we give further Alon–Füredi theorem provides degrees each variable are also taken into account. yields particular new proof Alon–Füredi. We then...

We give a new explicit construction of strong blocking sets in finite projective spaces using expander graphs and asymptotically good linear codes. Using the recently found equivalence between minimal codes, we first $\mathbb{F}_q$-linear codes length $n$ dimension $k$ such that is at most constant times $q k$. This solves one main open problems on

We prove that a minimal $t$-fold blocking set in finite projective plane of order $n$ has cardinality at most \[\frac{1}{2} n\sqrt{4tn - (3t + 1)(t 1)} \frac{1}{2} (t 1)n t.\] This is the first general upper bound on size sets planes and it generalizes classical result Bruen Thas sets. From proof directly follows if equality occurs this then every line intersects $S$ either $t$ points or $\frac{1}{2}(\sqrt{4tn t 1) 1$ points. use to show for prime power, can occur our exactly one following...

We construct and study a new near octagon of order $(2,10)$ which has its full automorphism group isomorphic to the $G_2(4):2$ contains $416$ copies Hall-Janko as subgeometries. Using this substructures we give geometric constructions $G_2(4)$-graph Suzuki graph, both are strongly regular graphs contained in tower. As subgeometry have discovered another octagon, whose is $(2,4)$.

Abstract We study the problem of determining minimum number $f(n,k,d)$ affine subspaces codimension $d$ that are required to cover all points $\mathbb{F}_2^n\setminus \{\vec{0}\}$ at least $k$ times while covering origin most $k - 1$ times. The case $k=1$ is a classic result Jamison, which was independently obtained by Brouwer and Schrijver for $d = . value $f(n,1,1)$ also follows from well-known theorem Alon Füredi about coverings finite grids in spaces over arbitrary fields. Here we...

A famous conjecture of Ryser states that every $r$-partite hypergraph has vertex cover number at most $r - 1$ times the matching number. In recent years, hypergraphs meeting this conjectured bound, known as $r$-Ryser hypergraphs, have been studied extensively. It was proved by Haxell, Narins, and Szabó all 3-Ryser with $\nu > are essentially obtained taking $\nu$ disjoint copies intersecting hypergraphs. Abu-Khazneh showed such a characterization is false for = 4$ giving computer generated...

We prove that s r ( K k ) = O 5 / 2 $s_r(K_k) O(k^5 r^{5/2})$ , where $s_r(K_k)$ is the Ramsey parameter introduced by Burr, Erdős and Lovász in 1976, which defined as smallest minimum degree of a graph G $G$ such any $r$ -colouring edges contains monochromatic $K_k$ whereas no proper subgraph has this property. The construction used our proof relies on group theoretic model generalised quadrangles Kantor 1980.

We give a computer-based proof for the non-existence of distance-$2$ ovoids in dual split Cayley hexagon $\mathsf{H}(4)^D$. Furthermore, we upper bounds on partial $\mathsf{H}(q)^D$ $q \in \{2, 4\}$.

We use $p$-rank bounds on partial ovoids and the classical Ramsey numbers to obtain various upper $m$-ovoids in finite polar spaces. These imply non-existence of for new families give a probabilistic construction large when $m$ grows linearly with rank space. In special case symplectic spaces over binary field, we show an equivalence between generalisation Oddtown theorem from extremal set theory that has been studied under name nearly $m$-orthogonal sets fields. constructions these thus...

A strong $s$-blocking set in a projective space is of points that intersects each codimension-$s$ subspace spanning the subspace. We present an explicit construction such sets $(k - 1)$-dimensional over $\mathbb{F}_q$ size $O_s(q^s k)$, which optimal up to constant factor depending on $s$. This also yields affine blocking $\mathbb{F}_q^k$ with respect codimension-$(s+1)$ subspaces, and $s$-minimal codes. Our approach motivated by recent Alon, Bishnoi, Das, Neri $1$-blocking sets, uses...

We give conditions under which the number of solutions a system polynomial equations over finite field F_q characteristic p is divisible by p. Our setup involves substitution t_i |-> f_i(t_i) for auxiliary polynomials f_1,...,f_n in F_q[t]. recover as special cases results Chevalley-Warning and Morlaye-Joly. Then we investigate higher p-adic divisibilities, proving result that recovers Ax-Katz Theorem. also consider p-weight degrees, recovering work Moreno-Moreno, Moreno-Castro Castro-Castro-Velez.

A well-known conjecture, often attributed to Ryser, states that the cover number of an r-partite r-uniform hypergraph is at most r−1 times larger than its matching number. Despite considerable effort, particularly in intersecting case, this conjecture remains wide open, motivating pursuit variants original conjecture. Recently, Bustamante and Stein and, independently, Király Tóthmérész considered problem under assumption t-intersecting, conjecturing τ(H) such a H r−t. In these papers, it was...