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In Neri et al. (2022) [20] , the authors extended the family of 2-dimensional F q 2 t -linear MRD codes recently found in Longobardi et al. (2023) [13]. Also, for t ≥ 5 they determined equivalence classes of the elements in this new family and provided the exact number of inequivalent codes in it. In this article, we complete the study of the equivalence issue removing the restriction t ≥ 5. Moreover, we prove that in the case when t = 4 , the linear sets of the projective line PG (1 , q 8) ensuing from codes in the relevant family, are not equivalent to any one known so far. [ABSTRACT FROM AUTHOR]

Group action on the projective space PG(3,q)is a method which can be used to construct some geometric objects for example cap. We constructed new caps in PG(3,13) of degrees 2, 3, 4, 7,14 and sizes 2, 4, 5, 7, 10, 14, 17, 20, 28, 34, 35, 68, 70, 85, 119, 140, 170, 238, 340, 476, 595, 1190. Then the incomplete caps are extended to complete caps. [ABSTRACT FROM AUTHOR]

Consider the n-dimensional projective space PG (n , q) over a finite field with q elements. A spread in PG (n , q) is a set of lines which partition the point set. A parallelism is a partition of the set of lines by spreads. A deficiency one parallelism is a partial parallelism with one spread less than the parallelism. A transitive deficiency one parallelism corresponds to a parallelism possessing an automorphism group which fixes one spread and is transitive on the remaining spreads. Such parallelisms have been considered in many papers. As a result, an infinite family of transitive deficiency one parallelisms of PG (n , q) has been constructed for odd q, and it has been proved that the deficiency spread of a transitive deficiency one parallelism must be regular, and its automorphism group should contain an elation subgroup of order q 2 . In the present paper we construct parallelisms of PG (3 , 7) invariant under an elation group of order 49 with some additional properties, and thus we succeed to obtain all (46) transitive deficiency one parallelisms of PG (3 , 7) . The three parallelisms from the known infinite family are among them. As a by-product, we also construct a much bigger number (55,022) of parallelisms which have the same spread structure, but are not transitive deficiency one. [ABSTRACT FROM AUTHOR]

Let PG (n , q) be the n -dimensional projective space over the finite field F q. A spread in PG (n , q) is a set of mutually skew lines which partition the point set. A parallelism is a partition of the set of lines by spreads. The classification of parallelisms in small finite projective spaces is of interest for problems from projective geometry, design theory, network coding, error-correcting codes, and cryptography. All parallelisms of PG (3 , 2) and PG (3 , 3) are known. Parallelisms of PG (3 , 4) which are invariant under automorphisms of odd prime orders have also been classified. The present paper contributes to the classification of parallelisms in PG (3 , 4) with automorphisms of even order. We focus on cyclic groups of order four and the group of order two generated by a Baer involution. We examine invariants of the parallelisms such as the full automorphism group, the type of spreads and questions of duality. The results given in this paper show that the number of parallelisms of PG (3 , 4) is at least 8675365. Some future directions of research are outlined. [ABSTRACT FROM AUTHOR]

In this paper, we study the connections between linear codes and projective geometries over finite fields. Each of these two topics is interesting by itself and has been subject to substantial research. In the last decade, a lot of progress has been made in both areas. We introduce some of the basic ideas and connections between finite projective spaces and coding theory. We begin by studying projective geometries, from this, we introduce a very interesting action in projective planes which lead to many other interesting areas of finite geometry, coordination of the plane. We will coordinate the lines using point coordination. Our focus then shifts to coding theory and in particular three-dimensional linear codes. The linear code Cs,t n, q of s-spaces and t-spaces in a projective space PG (n, q), q = pd, p prime, is defined as the vector space spanned over Fqby the rows of the incidence matrix of s-spaces and t-spaces. Three-dimensional code applied on the constructed projective model: Fano Plane, like a3-dimensional vector space over F2. The Fano plane, like a model, occurs in algebraic geometry and geometric algebra in a number of cases, constructing a link between such important mathematical concepts. There are given different ways of constructing the model, taking in consider that it is impossible to label the Fano plane in such a way that all or just five of its lines would be ordinary. [ABSTRACT FROM AUTHOR]

This research aims to give a splitting structure of the projective line over the finite field of order twenty-seven that can be found depending on the factors of the line order. Also, the line was partitioned by orbits using the companion matrix. Finally, we showed the number of projectively inequivalent k-ares on the conic v(X²1-X0X2₂) through the standard frame of the plane PG (2,27). [ABSTRACT FROM AUTHOR]

A class of scattered linearized polynomials covering infinitely many field extensions is exhibited. More precisely, the q-polynomial over F q 6 , q ≡ 1 (mod 4) described in Bartoli et al. (ARS Math Contemp 19:125–145, 2020) and Zanella and Zullo (Discrete Math 343:111800, 2020) is generalized for any even n ≥ 6 to an F q -linear automorphism ψ (x) of F q n of order n. Such ψ (x) and some functional powers of it are proved to be scattered. In particular, this provides new maximum scattered linear sets of the projective line PG (1 , q n) for n = 8 , 10 . The polynomials described in this paper lead to a new infinite family of MRD-codes in F q n × n with minimum distance n - 1 for any odd q if n ≡ 0 (mod 4) and any q ≡ 1 (mod 4) if n ≡ 2 (mod 4) . [ABSTRACT FROM AUTHOR]

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An m-cover of lines of a finite projective space PG (r , q) (of a finite polar space P ) is a set of lines L of PG (r , q) (of P ) such that every point of PG (r , q) (of P ) contains m lines of L , for some m. Embed PG (r , q) in PG (r , q 2) . Let L ¯ denote the set of points of PG (r , q 2) lying on the extended lines of L . An m-cover L of PG (r , q) is an (r - 2) -dual m-cover if there are two possibilities for the number of lines of L contained in an (r - 2) -space of PG (r , q) . Basing on this notion, we characterize m-covers L of PG (r , q) such that L ¯ is a two-character set of PG (r , q 2) . In particular, we show that if L is invariant under a Singer cyclic group of PG (r , q) then it is an (r - 2) -dual m-cover. Assuming that the lines of L are lines of a symplectic polar space W (r , q) (of an orthogonal polar space Q (r , q) of parabolic type), similarly to the projective case we introduce the notion of an (r - 2) -dual m-cover of symplectic type (of parabolic type). We prove that an m-cover L of W (r , q) (of Q (r , q) ) has this dual property if and only if L ¯ is a tight set of an Hermitian variety H (r , q 2) or of W (r , q 2) (of H (r , q 2) or of Q (r , q 2) ). We also provide some interesting examples of (4 n - 3) -dual m-covers of symplectic type of W (4 n - 1 , q) . [ABSTRACT FROM AUTHOR]

A k-arc in the projective space PG (n , q) is a set of k projective points such that no subcollection of n + 1 points is contained in a hyperplane. In this paper, we construct new 60-arcs and 110-arcs in PG (4 , q) that do not arise from rational or elliptic curves. We introduce computational methods that, when given a set P of projective points in the projective space of dimension n over an algebraic number field Q (ξ) , determines a complete list of primes p for which the reduction modulo p of P to the projective space PG (n , p h) may fail to be a k-arc. Using these methods, we prove that there are infinitely many primes p such that PG (4 , p) contains a PSL (2 , 11) -invariant 110-arc, where PSL (2 , 11) is given in one of its natural irreducible representations as a subgroup of PGL (5 , p) . Similarly, we show that there exist PSL (2 , 11) -invariant 110-arcs in PG (4 , p 2) and PSL (2 , 11) -invariant 60-arcs in PG (4 , p) for infinitely many primes p. [ABSTRACT FROM AUTHOR]

There are two known families of maximum scattered F q -linear sets in PG ( 1 , q t ) : the linear sets of pseudoregulus type and for t ≥ 4 the scattered linear sets found by Lunardon and Polverino. For t = 4 we show that these are the only maximum scattered F q -linear sets and we describe the orbits of these linear sets under the groups PGL ( 2 , q 4 ) and PΓL ( 2 , q 4 ) . [ABSTRACT FROM AUTHOR]

Let L be a linear set of pseudoregulus type in a line $$\ell $$ in $$\varSigma ^*={\mathrm {PG}}(t-1,q^t)$$ , $$t=5$$ or $$t>6$$ . We provide examples of q-order canonical subgeometries $$\varSigma _1,\, \varSigma _2 \subset \varSigma ^*$$ such that there is a $$(t-3)$$ -subspace $$\varGamma \subset \varSigma ^*\setminus (\varSigma _1 \cup \varSigma _2 \cup \ell )$$ with the property that for $$i=1,2$$ , L is the projection of $$\varSigma _i$$ from center $$\varGamma $$ and there exists no collineation $$\phi $$ of $$\varSigma ^*$$ such that $$\varGamma ^{\phi }=\varGamma $$ and $$\varSigma _1^{\phi }=\varSigma _2$$ . Condition (ii) given in Theorem 3 in Lavrauw and Van de Voorde (Des Codes Cryptogr 56:89-104, 2010) states the existence of a collineation between the projecting configurations (each of them consisting of a center and a subgeometry), which give rise by means of projections to two linear sets. It follows from our examples that this condition is not necessary for the equivalence of two linear sets as stated there. We characterize the linear sets for which the condition above is actually necessary. [ABSTRACT FROM AUTHOR]

Scattered linear sets of pseudoregulus type in PG ( 1 , q t ) have been defined and investigated in [19] . The aim of this paper is to continue such an investigation. Properties of a scattered linear set of pseudoregulus type, say L , are proved by means of three different ways to obtain L : (i) as projection of a q -order canonical subgeometry [20] , (ii) as a point set whose image under the field reduction map is the hypersurface of degree t in PG ( 2 t − 1 , q ) studied in [10] , (iii) as exterior splash, by the correspondence described in [15] . In particular, given a canonical subgeometry Σ of PG ( t − 1 , q t ) , necessary and sufficient conditions are given for the projection of Σ with center a ( t − 3 ) -subspace to be a linear set of pseudoregulus type. Furthermore, the q -order sublines are counted and geometrically described. [ABSTRACT FROM AUTHOR]

Let q be a power of a prime. Let X ⊆ PG(n, q) be the intersection of s quadric hypersurfaces. Assume n ≥ 6s. Here we prove that for all P,Q ∈ X there are lines D1,D2 ⊆ X with P ∈ D1, Q ∈ D2 and D1 ∩ D2 not empty.

Abstract: In this note, the existence of and linear codes is proven. These ternary codes are constructed from orbits of a projectivity group of , for . These new codes and their punctured subcodes improve the best known lower bounds on the largest possible minimum distance. [Copyright &y& Elsevier]

We construct an infinite family of transitive 42-arcs in $$\text{ PG}(3,q^{2})$$, with $$q=p^{n}\ge 29$$ and $$q\equiv 1\pmod {7}$$, under the action of the group $$\text{ PSL}(2,7)$$ in its representation as a subgroup of $$\text{ PGL}(4,q)$$. Further, we study the case $$q=29$$ in detail with computer assistance. For $$q=29$$ these 42-arcs turn out to be complete. [ABSTRACT FROM AUTHOR]

A fundamental question for simplicial complexes is to find the lowest dimensional Euclidean space in which they can be embedded. We investigate this question for order complexes of posets. We show that order complexes of thick geometric lattices as well as several classes of finite buildings, all of which are order complexes, are hard to embed. That means that such $$d$$ -dimensional complexes require $$(2d + 1)$$ -dimensional Euclidean space for an embedding. (This dimension is always sufficient for any $$d$$ -complex.) We develop a method to show non-embeddability for general order complexes of posets. [ABSTRACT FROM AUTHOR]

Let $${L = \mathbb{F}_{q^n}}$$ be a finite field and let $${F=\mathbb{F}_q}$$ be a subfield of L. Consider L as a vector space over F and the associated projective space that is isomorphic to PG( n − 1, q). The properties of the projective mapping induced by $${x \mapsto x^{-1}}$$ have been studied in Csajbók (Finite Fields Appl. 19:55-66, ), Faina et al. (Eur. J. Comb. 23:31-35, ), Havlicek (Abh. Math. Sem. Univ. Hamburg 53:266-275, ), Herzer (Abh. Math. Sem. Univ. Hamburg 55:211-228 , Handbook of Incidence Geometry. Buildings and Foundations. Elsevier, Amsterdam, ). The image of any line is a normal rational curve in some subspace. In this note a more detailed geometric description is achieved. Consequences are found related to mixed partitions of the projective spaces; in particular, it is proved that for any positive integer k, if q ≥ 2 − 1, then there are partitions of PG(2 − 1, q) in normal rational curves of degree 2 − 1. For smaller q the same construction gives partitions in ( q + 1)-tuples of independent points. [ABSTRACT FROM AUTHOR]

We prove that every possible k-cycle can be embedded into PG( n, q), for all n ≥ 3. [ABSTRACT FROM AUTHOR]

In this paper we reformulate the well-known result in the literature on the total number of lines of a finite projective space and so we give a detailed proof of the result. [ABSTRACT FROM AUTHOR]

Difference systems of sets (DSSs) are combinatorial structures introduced by Levenshtein in connection with code synchronization. In this paper, some recursive constructions of DSSs obtained from finite projective geometry are presented. As a consequence, new infinite families of optimal DSSs are obtained. [ABSTRACT FROM PUBLISHER]

Maximum scattered linear sets in PG (1 , q n) have been completely classified for n ≤ 4 , see Csajbók and Zanella (2018) [9] ; Lavrauw and Van de Voorde (2010) [10]. Here a wide class of linear sets in PG (1 , q 5) is studied which depends on two parameters. Conditions for the existence, in this class, of possible new maximum scattered linear sets in PG (1 , q 5) are exhibited. [ABSTRACT FROM AUTHOR]

The following result is well-known for finite projective spaces. The smallest cardinality of a set of points of PG( n, q) with the property that every s-subspace has a point in the set is ( q n+1- s - 1)/( q - 1). We solve in finite projective spaces PG( n, q) the following problem. Given integers s and b with 0 ≤ s ≤ n - 1 and 1 ≤ b ≤ ( q n+1- s - 1)/( q - 1), what is the smallest number of s-subspaces that must miss a set of b points. If d is the smallest integer such that b ≤ ( q d+1 - 1)/( q - 1), then we shall see that the smallest number is obtained only when the b points generate a subspace of dimension d. We then also determine the smallest number of s-subspaces that must miss a set of b points of PG( n, q) which do not lie together in a subspace of dimension d. The results are obtained by geometrical and combinatorial arguments that rely on a strong algebraic result for projective planes by T. Szőnyi and Z. Weiner. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present a construction of t-divisible designs (DDs) for t>3, because such DDs seem to be missing in the literature. To this end, tools such as finite projective spaces and their algebraic varieties are employed. More precisely, in a first step an abstract construction, called t-lifting, is developed. It starts from a set X containing a t-DD and a group G acting on X. Then several explicit examples are given, where X is a subset of PG(n,q) and G is a subgroup of GLn+1(q). In some cases X is obtained from a cone with a Veronesean or an h-sphere as its basis. In other examples, X arises from a projective embedding of a Witt design. As a result, for any integer t ≥ 2 infinitely many non-isomorphic t-DDs are found. [ABSTRACT FROM AUTHOR]

The following result is well-known for finite projective spaces. The smallest cardinality of a set of points of PG( n, q) with the property that every s-subspace has a point in the set is ( q n+1- s - 1)/( q - 1). We solve in finite projective spaces PG( n, q) the following problem. Given integers s and b with 0 ≤ s ≤ n - 1 and 1 ≤ b ≤ ( q n+1- s - 1)/( q - 1), what is the smallest number of s-subspaces that must miss a set of b points. If d is the smallest integer such that b ≤ ( q d+1 - 1)/( q - 1), then we shall see that the smallest number is obtained only when the b points generate a subspace of dimension d. We then also determine the smallest number of s-subspaces that must miss a set of b points of PG( n, q) which do not lie together in a subspace of dimension d. The results are obtained by geometrical and combinatorial arguments that rely on a strong algebraic result for projective planes by T. Szőnyi and Z. Weiner. [ABSTRACT FROM AUTHOR]

We provide a description of the configuration arising from intersection of two Hermitian surfaces in PG(3, q), provided that the linear system they generate contains at least a degenerate variety. [ABSTRACT FROM AUTHOR]

We study several interpolation problems for homogeneous polynomials over a finite field. [ABSTRACT FROM AUTHOR]

Abstract: The quadric Veronesean in PG is characterized as a -cap of class and type of PG by Ferri (q odd) and Thas and Van Maldeghem (q even). In this note we generalize this result slightly by proving that for a -cap of class and type of PG, the parameters m and n are uniquely determined and equal and , respectively. [Copyright &y& Elsevier]

In this article, it is shown that the largest partial linear complex of PG(3,q) that is not contained in a linear complex has q3+3 lines. We also show that such an inextendable partial linear complex of size q3+3 is projectively unique except in the case q=3. [Copyright &y& Elsevier]

Bruen proved that if A is a set of points in AG(n,q) which intersects every hyperplane in at least t points, then |A|⩾(n+t−1)(q−1)+1, leaving as an open question how good such bound is. Here we prove that, up to a trivial case, if t>((n−1)(q−1)+1)/2, then Bruen''s bound can be improved. If t is equal to the integer part of ((n−1)(q−1)+1)/2, then there are some examples which attain such a lower bound. Somehow, this suggests the following combinatorial characterization: if a set S of points in PG(3,q) meets every affine plane in at least q−1 points and is of minimum size with respect to this property, then S is a hyperbolic quadric. [Copyright &y& Elsevier]

A linearized polynomial f (x) ∈ F q n [ x ] is called scattered if for any y , z ∈ F q n , the condition z f (y) − y f (z) = 0 implies that y and z are F q -linearly dependent. In this paper two generalizations of the notion of a scattered linearized polynomial are provided and investigated. Let t be a nontrivial positive divisor of n. By weakening the property defining a scattered linearized polynomial, L- q t -partially scattered and R- q t -partially scattered linearized polynomials are introduced in such a way that the scattered linearized polynomials are precisely those which are both L- q t - and R- q t -partially scattered. Also, connections between partially scattered polynomials, linear sets and rank metric codes are exhibited. [ABSTRACT FROM AUTHOR]

In this paper we deal with authentication systems in which one key is used to authenticate many source states. We answer a related question on the cardinalities of the intersections of quadrics in PG (d,q). We first generalize a class of geometric authentication systems, which has been introduced by Beutelspacher, Tallini and Zanella4. The source states are the lines through a special point N of PG (d,q) (the d-dimensional projective space over GF (q)). The keys are some hypersurfaces which have N as a nucleus ( N is a nucleus of Σ if every line through N meets Σ in exactly one point). The message belonging to a source state ℓ and a key Σ is the unique point of intersection of the line ℓ with the hypersurface Σ. We give the values of s for which the constructed authentication systems have a security which is comparable to the best allowed by a theoretical bound. In case the hypersurfaces are quadrics, we give further results on the security. To this end, we determine the greatest cardinality for the intersections of the finite Veronese varieties with the projective subspaces of any given dimension. Finally, we discuss a possible implementation. [ABSTRACT FROM AUTHOR]