A proof of the linearity conjecture for k-blocking sets in PG(n, p3), p prime

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 sets in PG(n, p^3), p prime, p >=7, are Fp-linear, proving the linearity conjecture (see [7]) in the case PG(n, p3), p prime, p >= 7

On linear sets of minimum size

An $\mathbb{F}_q$-linear set of rank $k$ on a projective line $\mathrm{PG}(1,q^h)$, containing at least one point of weight one, has size at least $q^{k-1}+1$ (see [J. De Beule and G. Van De Voorde, The minimum size of a linear set, J. Comb. Theory, Ser: A 164 (2019), 109-124.]). The classical example of such a set is given by a club. In this paper, we construct a broad family of linear sets meeting this lower bound, where we are able to prescribe the weight of the heaviest point to any value between $k/2$ and $k-1$. Our construction extends the known examples of linear sets of size $q^{k-1}+1$ in $\mathrm{PG}(1,q^h)$ constructed for $k=h=4$ [G. Bonoli and O. Polverino, $\mathbb{F}_q$-Linear blocking sets in $\mathrm{PG}(2,q^4)$, Innov. Incidence Geom. 2 (2005), 35--56.] and $k=h$ in [G. Lunardon and O. Polverino. Blocking sets of size $q^t+q^{t-1}+1$. J. Comb. Theory, Ser: A 90 (2000), 148-158.]. We determine the weight distribution of the constructed linear sets and describe them as the projection of a subgeometry. For small $k$, we investigate whether all linear sets of size $q^{k-1}+1$ arise from our construction. Finally, we modify our construction to define linear sets of size $q^{k-1}+q^{k-2}+\ldots+q^{k-l}+1$ in $\mathrm{PG}(l,q)$. This leads to new infinite families of small minimal blocking sets which are not of R\'edei type

Large weight code words in projective space codes

AbstractRecently, a large number of results have appeared on the small weights of the (dual) linear codes arising from finite projective spaces. We now focus on the large weights of these linear codes. For q even, this study for the code Ck(n,q)⊥ reduces to the theory of minimal blocking sets with respect to the k-spaces of PG(n,q), odd-blocking the k-spaces. For q odd, in a lot of cases, the maximum weight of the code Ck(n,q)⊥ is equal to qn+⋯+q+1, but some unexpected exceptions arise to this result. In particular, the maximum weight of the code C1(n,3)⊥ turns out to be 3n+3n-1. In general, the problem of whether the maximum weight of the code Ck(n,q)⊥, with q=3h (h⩾1), is equal to qn+⋯+q+1, reduces to the problem of the existence of sets of points in PG(n,q) intersecting every k-space in 2(mod3) points

Strong blocking sets and minimal codes from expander graphs

A strong blocking set in a finite projective space is a set of points that intersects each hyperplane in a spanning set. We provide a new graph theoretic construction of such sets: combining constant-degree expanders with asymptotically good codes, we explicitly construct strong blocking sets in the $(k-1)$-dimensional projective space over $\mathbb{F}_q$ that have size $O( q k )$. Since strong blocking sets have recently been shown to be equivalent to minimal linear codes, our construction gives the first explicit construction of $\mathbb{F}_q$-linear minimal codes of length $n$ and dimension $k$, for every prime power $q$, for which $n = O (q k)$. This solves one of the main open problems on minimal codes.Comment: 20 page

On the linearity of higher-dimensional blocking sets

A small minimal k-blocking set B in PG(n,q), q = p(t), p prime, is a set of less than 3(q(k)+1)/2 points in PC(n,q), such that every (n - k)-dimensional space contains at least one point of B and such that no proper subset of B satisfies this property. The linearity conjecture states that all small minimal k-blocking sets in PG(n,q) are linear over a subfield F(pe) of F(q). Apart from a few cases, this conjecture is still open. In this paper, we show that to prove the linearity conjecture for k-blocking sets in PG(n,p(t)), with exponent e and p(e) >= 7, it is sufficient to prove it for one value of n that is at least 2k. Further more, we show that the linearity of small minimal blocking sets in PG(2,q) implies the linearity of small minimal k-blocking sets in PG(n,p(t)), with exponent epsilon, with p(e) >= t/epsilon + 11

All minimal [9,4]2[9,4]_{2}-codes are hyperbolic quadrics

An $[n,k]_{q}$-linear code is a $k$-dimensional subspace of $\mathbb{F}_{q}^{n}$. A codeword $c$ is minimal if its support does not contain the support of any codeword $c'\neq\lambda c$, $\lambda\in \mathbb{F}^*_{q}$. A linear code is minimal if all its codewords are minimal. $[n,k]_{q}$-minimal linear codes are in bijection with strong blocking sets of size $n$ in $PG(k-1,q)$ and a lower bound for the size of strong blocking set is given by $(k-1)(q+1)\leq n$. In this note we show that all strong blocking set of length 9 in $PG(3,2)$ are the hyperbolic quadrics $Q^{+}(3,2)$

Blocking sets of small size and colouring in finite affine planes

Let (S, L) be an either linear or semilinear space and X ⊂ S. Starting from X we define three types of colourings of the points of S. We characterize the Steiner systems S(2, k, ν) which have a colouring of the first type with X = {P}. By means of such colourings we construct blocking sets of small size in affine planes of order q. In particular, from the second and third type of colourings we get blocking sets B with |B| ≤ 2q − 2

Rings of geometries II

AbstractLinear spaces are investigated using the general theory of “Rings of Geometries I.” By defining geometries and ring structures in several different ways, formulae for linear spaces embedded in finite projective and affine planes are obtained. Several “fundamental theorems” of counting in finite projective planes are proved which show why configurations with at least three points per line and at least three lines through every point are important. These theorems are illustrated by finding the formulae for the number of k-arcs in a projective plane of order q for all k ⩽ 8 and also by finding a formula for the number of blocking sets. A quick proof that a projective plane of order 6 does not exist follows from the formula for the number of 7-arcs in such a plane