6 research outputs found
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
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