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In this paper, we solve a classical counting problem for non-degenerate quadratic forms defined on a vector space in odd characteristic; given a subspace $\pi$, we determine the number of non-singular subspaces that are trivially intersecting with $\pi$ and span a non-singular subspace with $\pi$. Lower bounds for the quantity of such pairs where $\pi$ is non-singular were first studied in `Glasby, Niemeyer, Praeger (Finite Fields Appl., 2022)', which was later improved for even-dimensional subspaces in `Glasby, Ihringer, Mattheus (Des. Codes Cryptogr., 2023)' and generalised in `Glasby, Niemeyer, Praeger (Linear Algebra Appl., 2022)'. The explicit formulae, which allow us to give the exact proportion and improve the known lower bounds were derived in the symplectic and Hermitian case in `De Boeck and Van de Voorde (Linear Algebra Appl. 2024)'. This paper deals with the more complicated quadratic case.

In this paper, we consider how probability theory can be used to determine the survival strategy in two of the ``Squid Game" and ``Squid Game: The Challenge" challenges: the Hopscotch and the Warships. We show how Hopscotch can be easily tackled with the knowledge of the binomial distribution, taught in introductory statistics courses, while Warships is a much more complex problem, which can be tackled at different levels., Comment: 10 pages, 5 figures

In this paper, we solve a classical counting problem for non-degenerate forms of symplectic and hermitian type defined on a vector space: given a subspace $\pi$, we find the number of non-singular subspaces that are trivially intersecting with $\pi$ and span a non-singular subspace with $\pi$. Lower bounds for the quantity of such pairs where $\pi$ is non-singular were first studied in ``Glasby, Niemeyer, Praeger (Finite Fields Appl., 2022)'', which was later improved in ``Glasby, Ihringer, Mattheus (Des. Codes Cryptogr., 2023)'' and generalised in ``Glasby, Niemeyer, Praeger (Linear Algebra Appl., 2022)''. In this paper, we derive explicit formulae, which allow us to give the exact proportion and improve the known lower bounds.

There are finitely many graphs with diameter $2$ and girth 5. What if the girth 5 assumption is relaxed? Apart from stars, are there finitely many triangle-free graphs with diameter $2$ and no $K_{2,3}$ subgraph? This question is related to the existence of triangle-free strongly regular graphs, but allowing for a range of co-degrees gives the question a more extremal flavour. More generally, for fixed $s$ and $t$, are there infinitely many twin-free triangle-free $K_{s,t}$-free graphs with diameter 2? This paper presents partial results regarding these questions, including computational results, potential Cayley-graph and probabilistic constructions.

We use techniques from algebraic and extremal combinatorics to derive upper bounds on the number of independent sets in several (hyper)graphs arising from finite geometry. In this way, we obtain asymptotically sharp upper bounds for partial ovoids and EKR-sets of flags in polar spaces, line spreads in $\mathrm{PG}(2r-1,q)$ and plane spreads in $\mathrm{PG}(5,q)$, and caps in $\mathrm{PG}(3,q)$. The latter result extends work due to Roche-Newton and Warren and Bhowmick and Roche-Newton. Finally, we investigate caps in $p$-random subsets of $\mathrm{PG}(r,q)$, which parallels recent work for arcs in projective planes by Bhowmick and Roche-Newton, and by Roche-Newton and Warren, and arcs in projective spaces by Chen, Liu, Nie and Zeng., Comment: 19 pages

This paper studies {\em strong blocking sets} in the $N$-dimensional finite projective space $\mathrm{PG}(N,q)$. We first show that certain unions of blocking sets cannot form strong blocking sets, which leads to a new lower bound on the size of a strong blocking set in $\mathrm{PG}(N,q)$. Our second main result shows that, for $q>\frac{2}{ln(2)}(N+1)$, there exists a subset of $2N-2$ lines of a Desarguesian line spread in $\mathrm{PG}(N,q)$, $N$ odd, in {\em higgledy-piggledy arrangement}; thus giving rise to a strong blocking set of size $(2N-2)(q+1)$.

In this paper, we characterise ovoidal cones by their intersection numbers. We first show that a set of points of $\mathrm{PG}(4,q)$ which intersects planes in $1$, $q+1$ or $2q+1$ points is either an ovoidal cone or a parabolic quadric, unless $q=3$, in which case also a sporadic example with automorphism group $M_{11}$ exists. We then show that a set of points of $\mathrm{PG}(4,q)$ which blocks all planes and intersects solids in $q+1$, $q^2+1$ or $q^2+q+1$ points is a plane or an ovoidal cone, and determine all examples that arise when the blocking condition is omitted., Comment: We were informed that Theorem 1.1 in the paper was already proved by Innamorati and Zuanni. Theorems 1.2 and 1.3 remain new results

It is known that a Bruen chain of the three-dimensional projective space $\mathrm{PG}(3,q)$ exists for every odd prime power $q$ at most $37$, except for $q=29$. It was shown by Cardinali et. al (2005) that Bruen chains do not exist for $41\le q\leq 49$. We develop a model, based on finite fields, which allows us to extend this result to $41\leqslant q \leqslant 97$, thereby adding more evidence to the conjecture that Bruen chains do not exist for $q>37$. Furthermore, we show that Bruen chains can be realised precisely as the $(q+1)/2$-cliques of a two related, yet distinct, undirected simple graphs.

In this paper, we investigate the Andr\'e/Bruck-Bose representation of certain $\mathbb{F}_q$-linear sets contained in a line of $\text{PG}(2,q^t)$. We show that scattered $\mathbb{F}_q$-linear sets of rank $3$ in $\text{PG}(1,q^3)$ correspond to particular hyperbolic quadrics and that $\mathbb{F}_q$-linear clubs in $\text{PG}(1,q^t)$ are linked to subspaces of a certain $2$-design based on normal rational curves; this design extends the notion of a circumscribed bundle of conics. Finally, we use these results to construct optimal higgledy-piggledy sets of planes in $\text{PG}(5,q)$., Comment: 18 pages

This paper addresses a number of problems concerning Buekenhout-Tits unitals in $PG(2,q^2)$, where $q = 2^{e+1}$ and $e \geq 1$. We show that all Buekenhout-Tits unitals are $PGL$-equivalent (addressing an open problem in [S. Barwick and G. L. Ebert. Unitals in projective planes. Springer Monographs in Mathematics. Springer, New York, 2008.]), explicitly describe their $P\Gamma L$-stabiliser (expanding Ebert's work in [G.L. Ebert. Buekenhout-Tits unitals. J. Algebraic. Combin. 6.2 (1997), 133-140], and show that lines meet the feet of points no on $\ell_\infty$ in at most four points. Finally, we show that feet of points not on $\ell_\infty$ are not always a $\{0,1,2,4\}$-set, in contrast to what happens for Buekenhout-Metz unitals [N. Abarz\'ua, R. Pomareda, and O. Vega. Feet in orthogonal-Buekenhout-Metz unitals. Adv. Geom. 18.2 (2018), 229-236].

The minimum weight of the code generated by the incidence matrix of points versus lines in a projective plane has been known for over 50 years. Surprisingly, finding the minimum weight of the dual code of projective planes of non-prime order is still an open problem, even in the Desarguesian case. In this paper, we focus on the case of projective planes of order $p^2$, where $p$ is prime, and we link the existence of small weight code words in the dual code to the existence of embedded subplanes and {\em antipodal planes}. In the Desarguesian case, we can exclude such code words by showing a more general result that no antipodal plane of order at least 3 can be embedded in a Desarguesian projective plane. Furthermore, we use combinatorial arguments to rule out the existence of code words in the dual code of points and lines of an arbitrary projective plane of order $p^2$, $p$ prime, of weight at most $2p^2-2p+4$ using more than two symbols. In particular, this leads to the result that the dual code of the Desarguesian projective plane $\mathrm{PG}(2,p^2)$, $p\geq 5$, has minimum weight at least $2p^2-2p+5$.

We provide a natural characterisation for the sets of elliptic and hyperbolic hyperplanes of the parabolic quadric Q(2n,q) when q is even. This characterisation is based on the number of elements of these sets through points and codimension 2 spaces and generalises [S. Barwick, A. Hui, and W-A. Jackson. Characterising elliptic solids of Q(4,q), q even. Discrete Math., 343 (6) (2020), 111857] and [S. Barwick, A. Hui, W-A. Jackson, and J. Schillewaert. Characterising hyperbolic solids of Q(4,q), q even. Des. Codes Cryptogr., 88 (1) (2020), 33--39.]., Comment: To appear in Finite Fields and their Applications

If an Fq-linear set LU in a projective space is defined by a vector subspace U which is linear over a proper superfield of Fq, then all of its points have weight at least 2. It is known that the converse of this statement holds for linear sets of rank h in PG(1,q^h) but for linear sets of rank k < h, the converse of this statement is in general no longer true. The first part of this paper studies the relation between the weights of points and the size of a linear set, and introduces the concept of the geometric field of linearity of a linear set. This notion will allow us to show the main theorem, stating that for particular linear sets without points of weight 1, the converse of the above statement still holds as long as we take the geometric field of linearity into account.

Quasi-polar spaces are sets of points having the same intersection numbers with respect to hyperplanes as classical polar spaces. Non-classical examples of quasi-quadrics have been constructed using a technique called pivoting [5]. We introduce a more general notion of pivoting, called switching, and also extend this notion to Hermitian polar spaces. The main result of this paper studies the switching technique in detail by showing that, for q >= 4, if we modify the points of a hyperplane of a polar space to create a quasi-polar space, the only thing that can be done is pivoting. The cases q = 2 and q = 3 play a special role for parabolic quadrics and are investigated in detail. Furthermore, we give a construction for quasi-polar spaces obtained from pivoting multiple times. Finally, we focus on the case of parabolic quadrics in even characteristic and determine under which hypotheses the existence of a nucleus (which was included in the definition given in [5]) is guaranteed.

In this paper, we study the weight distributions of $\mathbb{F}_q$-linear sets in $\mathrm{PG}(1,q^5)$. Our main theorem proves that a linear set $S$ of rank $5$, which is not scattered has the following weight distribution for its points with weight larger than 1: (i) one point of weight $4$ or $5$, (ii) one point of weight $3$ and $0$, $q$, $q^2$ points of weight two, (iii) $s$ points of weight $2$ where $s\in [q-2\sqrt{q}+1,q+2\sqrt{q}+1]\cup\{2q,2q+1,2q+2,3q,3q+1,q^2+1\}$. In particular, there are no $2$-clubs in $\mathrm{PG}(1,q^5)$.

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.

A Kakeya set $\mathcal{K}$ in an affine plane of order $q$ is the point set covered by a set $\mathcal{L}$ of $q+1$ pairwise non-parallel lines. Large Kakeya sets were studied by Dover and Mellinger; in [6] they showed that Kakeya sets with size at least $q^2-3q+9$ contain a large knot (a point of $\mathcal{K}$ lying on many lines of $\mathcal{L}$). In this paper, we improve on this result by showing that Kakeya set of size at least $\approx q^2-q\sqrt{q}+\frac{3}{2}q$ contain a large knot. Furthermore, we obtain a sharp result for planes of square order containing a Baer subplane., Comment: To appear in Advances in Geometry

Locally Repairable Codes (LRC's) based on generalised quadrangles were introduced by Pamies-Juarez, Hollmann and Oggier in \cite{PaHoOg2013}, and bounds on the repairability and availability were derived. In this paper, we determine the values of the repairability and availability of such LRC's for a large portion of the currently known generalised quadrangles. In order to do so, we determine the minimum weight of the codes of translation generalised quadrangles and characterise the codewords of minimum weight.

This paper deals with the following problem. Given a finite extension of fields $\mathbb{L}/\mathbb{K}$ and denoting the trace map from $\mathbb{L}$ to $\mathbb{K}$ by $\mathrm{Tr}$, for which elements $z$ in $\mathbb{L}$, and $a$, $b$ in $\mathbb{K}$, is it possible to write $z$ as a product $x\cdot y$, where $x,y\in \mathbb{L}$ with $\mathrm{Tr}(x)=a, \mathrm{Tr}(y)=b$? We solve most of these problems for finite fields, with a complete solution when the degree of the extension is at least $5$. We also have results for arbitrary fields and extensions of degrees $2,3$ or $4$. We then apply our results to the study of PN functions, semifields, irreducible polynomials with prescribed coefficients, and to a problem from finite geometry concerning the existence of certain disjoint linear sets.

In this paper, we study translation hyperovals in PG$(2,q^k)$. The main result of this paper characterises the point sets defined by translation hyperovals in the Andr\'e/Bruck-Bose representation. We show that the affine point sets of translation hyperovals in PG$(2,q^k)$ are precisely those that have a scattered $\mathbb{F}_2$-linear set of pseudoregulus type in PG$(2k-1,q)$ as set of directions. This correspondence is used to generalise the results of Barwick and Jackson who provided a characterisation for translation hyperovals in PG$(2,q^2)$.

In this paper we investigate connections between linear sets and subspaces of linear maps. We give a geometric interpretation of the results of [18, Section 5] on linear sets on a projective line. We extend this to linear sets in arbitrary dimension, giving the connection between two constructions for linear sets defined in [9]. Finally, we then exploit this connection by using the MacWilliams identities to obtain information about the possible weight distribution of a linear set of rank n on a projective line $PG(1, q^n)$.

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 and Silverman. In this note, we improve on this result by showing that a blocking set of an affine plane of order $q$, $q\geq 25$, contains at least $q+\lfloor\sqrt{q}\rfloor+3$ points.

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 subspace. We then use this result to find a lower bound on the number of points in an Fq- linear set of rank k in PG(2, q^n). In the case k = n, this confirms a conjecture by Sziklai in [9].

In this paper, we study KM-arcs in $PG(2, q)$, the Desarguesian projective plane of order $q$. A KM-arc A of type $t$ is a natural generalisation of a hyperoval: it is a set of $q + t$ points in $PG(2, q)$ such that every line of $PG(2,q)$ meets A in $0,2$ or $t$ points. We study a particular class of KM-arcs, namely, elation KM-arcs. These KM-arcs are highly symmetrical and moreover, many of the known examples are elation KM-arcs. We provide an algebraic framework and show that all elation KM-arcs of type $q/4$ in $PG(2,q)$ are translation KM-arcs. Using a result of [2], this concludes the classification problem for elation KM-arcs of type $q/4$. Furthermore, we construct for all $q = 2^h$, $h > 3$, an infinite family of elation KM-arcs of type $q/8$, and for $q = 2^h$, where $4, 6, 7 | h$ an infinite family of KM-arcs of type $q/16$. Both families contain new examples of KM-arcs.

The goal of this note is to create a sound framework for the interplay between field reduction for finite projective spaces, the general semilinear groups acting on the defining vector spaces and the projective semilinear groups. This approach makes it possible to reprove a result of Dye on the stabiliser in PGL of a Desarguesian spread in a more elementary way, and extend it to P{\Gamma}L(n, q). Moreover a result of Drudge [5] relating Singer cycles with Desarguesian spreads, as well as a result on subspreads (by Sheekey, Rottey and Van de Voorde [19]) are reproven in a similar elementary way. Finally, we try to use this approach to shed a light on Condition (A) of Csajbok and Zanella, introduced in the study of linear sets [4].

A {\em pseudo-arc} in $\mathrm{PG}(3n-1,q)$ is a set of $(n-1)$-spaces such that any three of them span the whole space. A pseudo-arc of size $q^n+1$ is a {\em pseudo-oval}. If a pseudo-oval $\mathcal{O}$ is obtained by applying field reduction to a conic in $\mathrm{PG}(2,q^n)$, then $\mathcal{O}$ is called a {\em pseudo-conic}. We first explain the connection of (pseudo-)arcs with Laguerre planes, orthogonal arrays and generalised quadrangles. In particular, we prove that the Ahrens-Szekeres GQ is obtained from a $q$-arc in $\mathrm{PG}(2,q)$ and we extend this construction to that of a GQ of order $(q^n-1,q^n+1)$ from a pseudo-arc of $\mathrm{PG}(3n-1,q)$ of size $q^n$. The main theorem of this paper shows that if $\mathcal{K}$ is a pseudo-arc in $\mathrm{PG}(3n-1,q)$, $q$ odd, of size larger than the size of the second largest complete arc in $\mathrm{PG}(2,q^n)$, where for one element $K_i$ of $\mathcal{K}$, the partial spread $\mathcal{S}=\{K_1,\ldots,K_{i-1},K_{i+1},\ldots,K_{s}\}/K_i$ extends to a Desarguesian spread of $\mathrm{PG}(2n-1,q)$, then $\mathcal{K}$ is contained in a pseudo-conic. The main result of \cite{Casse} also follows from this theorem.

We present a construction for minimal blocking sets with respect to $(k-1)$-spaces in $\mathrm{PG}(n-1,q^t)$, the $(n-1)$-dimensional projective space over the finite field $\mathbb{F}_{q^t}$ of order $q^t$. The construction relies on the use of blocking cones in the {\em field reduced} representation of $\mathrm{PG}(n-1,q^t)$, extending the well-known construction of linear blocking sets. This construction is inspired by the construction for minimal blocking sets with respect to the hyperplanes by Mazzocca, Polverino and Storme ({\em the MPS-construction}); we show that for a suitable choice of the blocking cone over a planar blocking set, we obtain larger blocking sets than the ones obtained from planar blocking sets in \cite{pol}. Furthermore we show that every minimal blocking set with respect to the hyperplanes in $\mathrm{PG}PG(n-1,q^t)$ can be obtained by applying field reduction to a minimal blocking set with respect to $(nt-t-1)$-spaces in $\mathrm{PG}(nt-1,q)$. We end by relating these constructions to the linearity conjecture for small minimal blocking sets. We show that if a small minimal blocking set is constructed from the MPS-construction, it is of R\'edei-type whereas a small minimal blocking set arises from our cone construction if and only if it is linear.

In this paper, we study KM-arcs of type t, i.e. point sets of size q + t in PG(2, q) such that every line contains 0, 2 or t of its points. We use field reduction to give a different point of view on the class of translation arcs. Starting from a particular F2-linear set, called an i-club, we reconstruct the projective triads, the translation hyperovals as well as the translation arcs constructed by Korchmaros-Mazzocca, Gacs-Weiner and Limbupasiriporn. We show the KM-arcs of type q/4 recently constructed by Vandendriessche are translation arcs and fit in this family. Finally, we construct a family of KM-arcs of type q/4. We show that this family, apart from new examples that are not translation KM-arcs, contains all translation KM-arcs of type q/4.

We show that a unital $U$ in $\mathrm{PG}(2,q^2)$ containing a point $P$, such that at least $q^2-\epsilon$ of the secant lines through $P$ intersect $U$ in a Baer subline, is an ovoidal Buekenhout-Metz unital (where $\epsilon\approx 2q$ for $q$ even and $\epsilon\approx q^{3/2}/2$ for $q$ odd).

In this note, we use the theory of Desarguesian spreads to investigate good eggs. Thas showed that an egg in $\mathrm{PG}(4n-1, q)$, $q$ odd, with two good elements is elementary. By a short combinatorial argument, we show that a similar statement holds for large pseudo-caps, in odd and even characteristic. As a corollary, this improves and extends the result of Thas, Thas and Van Maldeghem (2006) where one needs at least 4 good elements of an egg in even characteristic to obtain the same conclusion. We rephrase this corollary to obtain a characterisation of the generalised quadrangle $T_3(\mathcal{O})$ of Tits. Lavrauw (2005) characterises elementary eggs in odd characteristic as those good eggs containing a space that contains at least 5 elements of the egg, but not the good element. We provide an adaptation of this characterisation for weak eggs in odd and even characteristic. As a corollary, we obtain a direct geometric proof for the theorem of Lavrauw.

We consider the Andr\'e/Bruck-Bose representation of the projective plane $\mathrm{PG}(2,q^n)$ in $\mathrm{PG}(2n,q)$. We investigate the representation of $\mathbb{F}_{q^k}$-sublines and $\mathbb{F}_{q^k}$-subplanes of $\mathrm{PG}(2,q^n)$, extending the results for $n=3$ of \cite{BarJack2} and correcting the general result of \cite{BarJack1}. We characterise the representation of $\mathbb{F}_{q^k}$-sublines tangent to or contained in the line at infinity, $\mathbb{F}_q$-sublines external to the line at infinity, $\mathbb{F}_q$-subplanes tangent to and $\mathbb{F}_{q^k}$-subplanes secant to the line at infinity.

Pseudo-arcs are the higher dimensional analogues of arcs in a projective plane: a pseudo-arc is a set $\mathcal{A}$ of $(n-1)$-spaces in $\mathrm{PG}(3n-1,q)$ such that any three span the whole space. Pseudo-arcs of size $q^n+1$ are called pseudo-ovals, while pseudo-arcs of size $q^n+2$ are called pseudo-hyperovals. A pseudo-arc is called elementary if it arises from applying field reduction to an arc in $\mathrm{PG}(2,q^n)$. We explain the connection between dual pseudo-ovals and elation Laguerre planes and show that an elation Laguerre plane is ovoidal if and only if it arises from an elementary dual pseudo-oval. The main theorem of this paper shows that a pseudo-(hyper)oval in $\mathrm{PG}(3n-1,q)$, where $q$ is even and $n$ is prime, such that every element induces a Desarguesian spread, is elementary. As a corollary, we give a characterisation of certain ovoidal Laguerre planes in terms of the derived affine planes.

The linear representation $T_n^*(\mathcal{K})$ of a point set $\mathcal{K}$ in a hyperplane of $\mathrm{PG}(n+1,q)$ is a point-line geometry embedded in this projective space. In this paper, we will determine the isomorphisms between two linear representations $T_n^*(\mathcal{K})$ and $T_n^*(\mathcal{K}')$, under a few conditions on $\mathcal{K}$ and $\mathcal{K}'$. First, we prove that an isomorphism between $T_n^*(\mathcal{K})$ and $T_n^*(\mathcal{K}')$ is induced by an isomorphism between the two linear representations $T_n^*(\overline{\mathcal{K}})$ and $T_n^*(\overline{\mathcal{K}'})$ of their closures $\overline {\mathcal{K}}$ and $\overline{\mathcal{K}'}$. This allows us to focus on the automorphism group of a linear representation $T_n^*(\mathcal{S})$ of a subgeometry $\mathcal{S}\cong\mathrm{PG}(n,q)$ embedded in a hyperplane of the projective space $\mathrm{PG}(n+1,q^t)$. To this end we introduce a geometry $X(n,t,q)$ and determine its automorphism group. The geometry $X(n,t,q)$ is a straightforward generalization of $H_{q}^{n+2}$ which is known to be isomorphic to the linear representation of a Baer subgeometry. By providing an elegant algebraic description of $X(n,t,q)$ as a coset geometry we extend this result and prove that $X(n,t,q)$ and $T_n^*(\mathcal{S})$ are isomorphic. Finally, we compare the full automorphism group of $T^*_n(\mathcal{S})$ with the "natural" group of automorphisms that is induced by the collineation group of its ambient space.

Based on the simple and well understood concept of subfields in a finite field, the technique called `field reduction' has proved to be a very useful and powerful tool in finite geometry. In this paper we elaborate on this technique. Field reduction for projective and polar spaces is formalized and the links with Desarguesian spreads and linear sets are explained in detail. Recent results and some fundamental ques- tions about linear sets and scattered spaces are studied. The relevance of field reduction is illustrated by discussing applications to blocking sets and semifields.

We give a general construction leading to different non-isomorphic families $\Gamma_{n,q}(\K)$ of connected $q$-regular semisymmetric graphs of order $2q^{n+1}$ embedded in $\PG(n+1,q)$, for a prime power $q=p^h$, using the linear representation of a particular point set $\K$ of size $q$ contained in a hyperplane of $\PG(n+1,q)$. We show that, when $\K$ is a normal rational curve with one point removed, the graphs $\Gamma_{n,q}(\K)$ are isomorphic to the graphs constructed for $q$ prime in [9] and to the graphs constructed for $q=p^h$ in [20]. These graphs were known to be semisymmetric but their full automorphism group was up to now unknown. For $q\geq n+3$ or $q=p=n+2$, $n\geq 2$, we obtain their full automorphism group from our construction by showing that, for an arc $\K$, every automorphism of $\Gamma_{n,q}(\K)$ is induced by a collineation of the ambient space $\PG(n+1,q)$. We also give some other examples of semisymmetric graphs $\Gamma_{n,q}(\K)$ for which not every automorphism is induced by a collineation of their ambient space.

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 sets in PG(n,p^t), p prime, p>5t-11, spanning a (t-1)-dimensional space, are F_p-linear, hence confirming the linearity conjecture for blocking sets in this particular case.

In this paper, we show that a set of q+a hyperplanes, q>13, a<(q-10)/4, that does not cover PG(n,q), does not cover at least q^(n-1)-aq^(n-2) points, and show that this lower bound is sharp. If the number of non- covered points is at most q^(n-1), then we show that all non-covered points are contained in one hyperplane. Finally, using a recent result of Blokhuis, Brouwer, and Szonyi [3], we remark that the bound on a for which these results are valid can be improved to a<(q-2)/3 and that this upper bound on a is sharp

In this paper, we study the isomorphism problem for linear representations. A linear representation Tn*(K) of a point set K is a point-line geometry, embedded in a projective space PG(n+1,q), where K is contained in a hyperplane. We put constraints on K which ensure that every automorphism of Tn*(K) is induced by a collineation of the ambient projective space. This allows us to show that, under certain conditions, two linear representations Tn*(K) and Tn*(K') are isomorphic if and only if the point sets K and K' are PGammaL-equivalent. We also deal with the slightly more general problem of isomorphic incidence graphs of linear representations. In the last part of this paper, we give an explicit description of the group of automorphisms of Tn*(K) that are induced by collineations of PG(n+1,q)., Comment: 14 pages