Related papers: On cutting blocking sets and their codes
It is well know that the theory of minimal blocking sets is studied by several author. Another theory which is also studied by a large number of researchers is the theory of hyperplane arrangements. We can remark that the affine space…
Motivated by a question of Erd\H{o}s on blocking sets in a projective plane that intersect every line only a few times, several authors have used unions of algebraic curves to construct such sets in $\mathbb{P}^2(\mathbb{F}_q)$. In this…
A $\varrho$-saturating set of $\text{PG}(N,q)$ is a point set $\mathcal{S}$ such that any point of $\text{PG}(N,q)$ lies in a subspace of dimension at most $\varrho$ spanned by points of $\mathcal{S}$. It is generally known that a…
Let $\Pi_q$ be an arbitrary finite projective plane of order $q$. A subset $S$ of its points is called saturating if any point outside $S$ is collinear with a pair of points from $S$. Applying probabilistic tools we improve the upper bound…
We develop three approaches of combinatorial flavour to study the structure of minimal codes and cutting blocking sets in finite geometry, each of which has a particular application. The first approach uses techniques from algebraic…
In this paper, we show that a small minimal k-blocking set in PG(n, q3), q = p^h, h >= 1, p prime, p >=7, intersecting every (n-k)-space in 1 (mod q) points, is linear. As a corollary, this result shows that all small minimal k-blocking…
A strong $s$-blocking set in a projective space is a set of points that intersects each codimension-$s$ subspace in a spanning set of the subspace. We present an explicit construction of such sets in a $(k - 1)$-dimensional projective space…
Strong blocking sets, introduced first in 2011 in connection with saturating sets, have recently gained a lot of attention due to their correspondence with minimal codes. In this paper, we dig into the geometry of the concatenation method,…
We show that the metric dimension of a finite projective plane of order $q\geq 23$ is $4q-4$, and describe all resolving sets of that size. Let $\tau_2$ denote the size of the smallest double blocking set in $\mathrm{PG}(2,q)$, the…
We prove new upper bounds on the smallest size of affine blocking sets, that is, sets of points in a finite affine space that intersect every affine subspace of a fixed codimension. We show an equivalence between affine blocking sets with…
We describe small dominating sets of the incidence graphs of finite projective planes by establishing a stability result which shows that dominating sets are strongly related to blocking and covering sets. Our main result states that if a…
Saturating sets are combinatorial objects in projective spaces over finite fields that have been intensively investigated in the last three decades. They are related to the so-called covering problem of codes in the Hamming metric. In this…
Subspace codes are collections of subspaces of a projective space such that any two subspaces satisfy a pairwise minimum distance criterion. Recent results have shown that it is possible to construct optimal $(5,3)$ subspace codes from…
An untouchable set in a projective plane is a set of points such that no line of the plane meets the set in exactly one point. Recently, H\'eger and Nagy (Avoiding Secants of Given Size in Finite Projective Planes, J. Combin. Des.…
Minimal rank-metric codes or, equivalently, linear cutting blocking sets are characterized in terms of the second generalized rank weight, via their connection with evasiveness properties of the associated $q$-system. Using this result, we…
A blocking set in an affine plane is a set of points $B$ such that every line contains at least one point of $B$. The best known lower bound for blocking sets in arbitrary (non-desarguesian) affine planes was derived in the 1980's by Bruen…
In this paper, we study the p-ary linear code C(PG(n, q)), q = p^h, p prime, h >= 1, generated by the incidence matrix of points and hyperplanes of a Desarguesian projective space PG(n, q), and its dual code. We link the codewords of small…
In this paper, we first determine the minimum possible size of an Fq-linear set of rank k in PG(1, q^n). We obtain this result by relating it to the number of directions determined by a linearized polynomial whose domain is restricted to a…
In a projective plane $\Pi _{q}$ (not necessarily Desarguesian) of order $q,$ a point subset $S$ is saturating (or dense) if any point of $\Pi _{q}\setminus S$ is collinear with two points in$~S$. Using probabilistic methods, the following…
In this paper, we show that a small minimal blocking set with exponent e in PG(n,p^t), p prime, spanning a (t/e-1)-dimensional space, is an F_p^e-linear set, provided that p>5(t/e)-11. As a corollary, we get that all small minimal blocking…