Long binary linear codes and large caps in projective space

Finite Fields and Their Applications, 2018

In [2] and [19] are presented the first two families of maximum scattered F q-linear sets of the projective line PG(1, q n). More recently in [23] and in [5], new examples of maximum scattered F q-subspaces of V (2, q n) have been constructed, but the equivalence problem of the corresponding linear sets is left open. Here we show that the F q-linear sets presented in [23] and in [5], for n = 6, 8, are new. Also, for q odd, q ≡ ±1, 0 (mod 5), we present new examples of maximum scattered F q-linear sets in PG(1, q 6), arising from trinomial polynomials, which define new F q-linear MRD-codes of F 6×6 q with dimension 12, minimum distance 5 and middle nucleus (or left idealiser) isomorphic to F q 6 .

Designs, Codes and Cryptography, 2010

Linear sets generalise the concept of subgeometries in a projective space. They have many applications in finite geometry. In this paper we address two problems for linear sets: the equivalence problem and the intersection problem. We consider linear sets as quotient geometries and determine the exact conditions for two linear sets to be equivalent. This is then used to determine in which cases all linear sets of rank 3 of the same size on a projective line are (projectively) equivalent. In [3], the intersection problem for subgeometries of PG(n, q) is solved. The intersection of linear sets is much more difficult. We determine the intersection of a subline PG(1, q) with a linear set in PG(1, q h ) and investigate the existence of irregular sublines, contained in a linear set. We also derive an upper bound, which is sharp for odd q, on the size of the intersection of two different linear sets of rank 3 in PG(1, q h ).

Advances in Geometry, 2012

In [10], it was shown that small t-fold (n − k)-blocking sets in PG(n, q), q = p h , p prime, h ≥ 1, intersect every k-dimensional space in t (mod p) points. We characterize in this article all t-fold (n − k)-blocking sets in PG(n, q), q square, q ≥ 661, t < c p q 1/6 /2, |B| < tq n−k + 2tq n−k−1 √ q, intersecting every k-dimensional space in t (mod √ q) points. * The third author is grateful for the partial support of OTKA T049662, T067867 and Bolyai grants. In the theory of 1-fold planar blocking sets, 1 (mod p) results for small 1-fold planar blocking sets play an important role. Definition 1.3 A blocking set of PG(2, q) is called small when it has less than 3(q + 1)/2 points. If q = p h , p prime, h ≥ 1, the exponent e of the minimal blocking set B of PG(2, q) is the maximal integer e such that every line intersects B in 1 (mod p e) points. Theorem 1.4 Let B be a small minimal 1-fold blocking set in PG(2, q), q = p h , p prime, h ≥ 1. Then B intersects every line in 1 (mod p) points, so for the exponent e of B, we have 1 ≤ e ≤ h. (Szőnyi [18]) In fact, this exponent e is a divisor of h. (Sziklai [17]) This result was extended by Szőnyi and Weiner [19] to 1-fold (n − k)blocking sets in PG(n, q). Definition 1.5 A 1-fold (n−k)-blocking set of PG(n, q) is called small when it has less than 3(q n−k + 1)/2 points. If q = p h , p prime, h ≥ 1, the exponent e of the minimal 1-fold (n − k)blocking set B is the maximal integer e such that every hyperplane intersects B in 1 (mod p e) points. A most interesting question of the theory of blocking sets is to classify the small blocking sets. A natural construction (blocking the k-subspaces of PG(n, q)) is a subgeometry PG(h(n − k)/e, p e), if it exists (recall q = p h , so 1 ≤ e ≤ h and e|h).

Journal of Statistical Planning and Inference, 1996

A proper non-empty subset C of the points of a linear space S = (P; L) is called line-closed if any two intersecting lines of S , each meeting C at least twice, have their intersection in C. We show that when every line has k points and every point lies on r lines the maximum size for such sets is r + k ? 2. In addition it is shown that this cannot happen for projective spaces PG(n; q) unless q = 2, nor can it be obtained for a ne spaces AG(n; q) unless n = 2 and q = 3. However, for all odd values of r there exist Steiner triple systems having such maximum line-closed subsets.

The set of all subspaces of F n q is denoted by Pq(n). The subspace distance dS(X, Y ) = dim(X) + dim(Y ) − 2 dim(X ∩ Y ) defined on Pq(n) turns it into a natural coding space for error correction in random network coding.

Designs, Codes and Cryptography, 2008

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 weight of this code to blocking sets with respect to lines in PG(n,q) and we exclude all possible codewords arising from small linear blocking sets. We also look at the dual code of C(PG(n,q)) and we prove that finding the minimum weight of the dual code can be reduced to finding the minimum weight of the dual code of points and lines in PG(2,q). We present an improved upper bound on this minimum weight and we show that we can drop the divisibility condition on the weight of the codewords in Sachar’s lower bound (Geom Dedicata 8:407–415, 1979).

International Electronic Journal of Geometry

This note concerns some arrangements of lines in P N (C) and the condition under which there exists a hyperplane intersecting transversely every line of the given arrangement at a unique point.

Discrete Mathematics, 1977

Discrete Mathematics, 2003

A famous result of de Bruijn and Erdős (Indag. Math. 10 (1948) 421-423) states that a ÿnite linear space has at least as many lines as points, with equality only if it is a projective plane or a near-pencil. This result led to the problem of characterizing ÿnite linear spaces for which the di erence between the number b of lines and the number v of points is assigned. In this paper ÿnite linear spaces with b − v 6 m, m being the minimum number of lines on a point, are characterized.

Monatshefte f�r Mathematik, 1990

We call a convex subset N of a convex d-polytope P c E d a k-nucleus of P if N meets every k-face of P, where 0 < k < d. We note that P has disjoint k-nuclei if and only if there exists a hyperplane in E d which bisects the (relative) interior of every k-face of P, and that this is possible only if/~-/~< k ~< d-1.