9 research outputs found
Resolving sets for higher dimensional projective spaces
Lower and upper bounds on the size of resolving sets for the point-hyperplane incidence graph of the finite projective space PG(n, q) are presented
All minimal -codes are hyperbolic quadrics
An -linear code is a -dimensional subspace of
. A codeword is minimal if its support does not contain
the support of any codeword , . A
linear code is minimal if all its codewords are minimal. -minimal
linear codes are in bijection with strong blocking sets of size in
and a lower bound for the size of strong blocking set is given by
. In this note we show that all strong blocking set of length
9 in are the hyperbolic quadrics
Higgledy-piggledy sets in projective spaces of small dimension
This work focuses on higgledy-piggledy sets of -subspaces in
, i.e. sets of projective subspaces that are 'well-spread-out'.
More precisely, the set of intersection points of these -subspaces with any
-subspace of spans itself. We
highlight three methods to construct small higgledy-piggledy sets of
-subspaces and discuss, for , 'optimal' sets that cover the
smallest possible number of points. Furthermore, we investigate small
non-trivial higgledy-piggledy sets in , . Our main
result is the existence of six lines of in higgledy-piggledy
arrangement, two of which intersect. Exploiting the construction methods
mentioned above, we also show the existence of six planes of
in higgledy-piggledy arrangement, two of which maximally intersect, as well as
the existence of two higgledy-piggledy sets in consisting of
eight planes and seven solids, respectively. Finally, we translate these
geometrical results to a coding- and graph-theoretical context.Comment: [v1] 21 pages, 1 figure [v2] 21 pages, 1 figure: corrected minor
details, updated bibliograph
Small Strong Blocking Sets by Concatenation
Strong blocking sets and their counterparts, minimal codes, attracted lots of
attention in the last years. Combining the concatenating construction of codes
with a geometric insight into the minimality condition, we explicitly provide
infinite families of small strong blocking sets, whose size is linear in the
dimension of the ambient projective spaces. As a byproduct, small saturating
sets are obtained.Comment: 16 page
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
-dimensional projective space over that have size . Since strong blocking sets have recently been shown to be equivalent to
minimal linear codes, our construction gives the first explicit construction of
-linear minimal codes of length and dimension , for every
prime power , for which . This solves one of the main open
problems on minimal codes.Comment: 20 page
On subspace designs
Guruswami and Xing introduced subspace designs in 2013 to give the first
construction of positive rate rank metric codes list-decodable beyond half the
distance. In this paper we provide bounds involving the parameters of a
subspace design, showing they are tight via explicit constructions. We point
out a connection with sum-rank metric codes, dealing with optimal codes and
minimal codes with respect to this metric. Applications to two-intersection
sets with respect to hyperplanes, two-weight codes, cutting blocking sets and
lossless dimension expanders are also provided