We consider the problem of classifying the lines of the projective 3-space PG(3,q) over a finite field Fq into orbits of the group PGL2(q) of linear symmetries of the twisted cubic C . The problem has been solved in literature in characteristic different from 3, and in this work, we solve the problem in characteristic 3. We reduce this problem to another problem, which is the classification of binary quartic forms into PGL2(q)-orbits. We first solve the latter problem and use to solve the former problem. We also obtain the point-line and the line-plane incidence structures of the point, line, and plane orbits.

The problem studied in this work is to determine the higher weight spectra of the Projective Reed–Muller codes associated to the Veronese 3-fold V in PG(9, q), which is the image of the quadratic Veronese embedding of PG(3, q) in PG(9, q). We reduce the problem to the following combinatorial problem in finite geometry: For each subset S of V, determine the dimension of the linear subspace of PG(9, q) generated by S. We develop a systematic method to solve the latter problem. We implement the method for q = 3, and use it to obtain the higher weight spectra of the associated code. The case of a general finite field Fqwill be treated in a future work.

Consider a hyperbolic or an elliptic quadric Qϵ (3, q), ϵ ∈ {+, −}, in PG(3, q) and let Eϵ denote the set of all lines of PG(3, q) that are external with respect to Qϵ (3, q). If π is a (tangent or secant) plane of PG(3, q), then π Qϵ (3, q) is an Eϵ-blocking set which is minimal, except when (q, ϵ) = (2, +) and π is a secant plane. In this way, we obtain two families of (minimal) blocking sets of sizes mϵ1 and mϵ2, with (m+1, m+2) = (q2 − q, q2 ) and (m−1, m−2) = (q2, q2 + q), where those of size mϵ1 are also the minimum size Eϵ-blocking sets. Motivated by the search for new (families of ) minimal Eϵ-blocking sets with sizes in the open interval ]mϵ1, mϵ2[, we determine here all Eϵ-blocking sets of size mϵ1 + 1 in PG(3, q) in the case q ∈ {2, 3}.

For a set L of lines of PG(n,q), a set X of points of PG(n,q) is called an L-blocking set if each line of L contains at least one point of X. Consider a possibly singular quadric Q of PG(n,q) and denote by S (respectively, T) the set of all lines of PG(n,q) meeting Q in 2 (respectively, 1 or q+1) points. For L∈{S,T∪S}, we find the minimal cardinality of an L-blocking set of PG(n,q) and determine all L-blocking sets of that minimal cardinality.

Let Q−(3, q) be an elliptic quadric and Q+(3, q) a hyperbolic quadric in PG(3, q). For ɛ ∈ {−, +}, let Tɛ denote the set of all tangent lines of PG(3, q) with respect to Qɛ(3, q). If k is the minimum size of a Tɛ-blocking set in PG(3, q), then it is known that q2 +1 ≤ k ≤ q2 +q. For an odd prime q, we prove that there are no T+-blocking sets of size q2 + 1 and that the quadric Q−(3, q) is the only T−-blocking set of size q2 + 1 in PG(3, q). When q = 3, we show with the aid of a computer that there are no minimal T−-blocking sets of size 11 and that, up to isomorphism, there are eight minimal T−-blocking sets of size 12 in PG(3, 3). We also provide geometrical constructions for these eight mutually nonisomorphic minimal T−-blocking sets of size 12.

In [9], two of us classified line sets in PG(3,q), q odd, that satisfy a certain list of properties. It was shown there that if q≥7, then each such line set is either the set of secant lines with respect to a hyperbolic quadric of PG(3,q) or belongs to a certain “hypothetical family” of line sets (for which no examples were known in [9]). In the present paper, we achieve two goals. On the one hand, we extend the mentioned classification result to all odd prime powers q. On the other hand, we study the hypothetical family of line sets and show that they are related to quadratic sets of the Klein quadric. This will allow us to show that such line sets exist for every odd prime power q.

Consider an elliptic quadric Q-(3 , q) in PG (3 , q). Let E and T denote the set of all lines of PG (3 , q) which meet Q-(3 , q) in 0 and 1 point, respectively. In this paper, we characterize the minimum size (T∪ E) -blocking sets and give a different proof for the characterization of minimum size E-blocking sets in PG (3 , q) which works for all q. We also discuss whether the main results of this paper (Theorems 1.6 and 1.7) can be extended to ovoids in PG (3 , q).

Consider a quadratic cone K in the 3-dimensional projective space PG(3,q) over a finite field of order q, where q is a prime power. Let E (respectively, T, S) denote the set of all lines of PG(3,q) that are external (respectively, tangent, secant) with respect to K. We characterize the minimum size blocking sets in PG(3,q) with respect to the line set A, where A is one of E, T, S, E∪T, E∪S and T∪S.

We give a combinatorial characterization of the family of lines of PG(3,q), q≥7 odd, which meet a hyperbolic quadric in two points (the so called secant lines) using their intersection properties with the points and planes of PG(3,q).

For a given nonempty subset L of the line set of PG(3, q), a set X of points of PG(3, q) is called an L-blocking set if each line in L contains at least one point of X. Consider an elliptic quadric Q−(3, q) in PG(3, q). Let E (respectively, T, S) denote the set of all lines of PG(3, q) which meet Q−(3, q) in 0 (respectively, 1, 2) points. In this paper, we characterize the minimum size L-blocking sets in PG(3, q), where L is one of the line sets S, E ∪ S, and T ∪ S.