Resolving sets for higher dimensional projective spaces

Abstract

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.

On resolving sets in the point-line incidence graph of PG(n, q)

“…In the former case resolving sets are connected with lines in a higgledypiggledy arrangement which were investigated by Fancsali and Sziklai [9]. Their results were recently improved by the authors of this paper [5]. The latter case is studied in the present paper.…”

Small Strong Blocking Sets by Concatenation

“…Note that the above result cannot be used to obtain strong blocking sets for large dimensions. There are stronger results for small dimensions: see [16,19] for k = 3, 4, [6] for k = 5, and [5] for k = 6.…”

“…Another possibility is to obtain short minimal codes by concatenating MDS codes over F q 3 with short minimal codes of dimension 3 over F q (see [8] for an upper bound on their minimal lengths). We get minimal codes shorter than the upper bound of Theorem 1.6 for the following parameters: [18,6] Finally, let us remark that we use inner codes corresponding to the tetrahedron (see [1, Theorem 5.3.]) or some shorter ones constructed in [2], whose length is quadratic in the dimension.…”

“…In the binary case, we know (see [1]) that the shortest minimal codes in dimensions ≤ 5 have parameters [3,2,2] 2 , [6, 3, 3] 2 , [9,4,4] 2 , [13,5,5] 2 . In the ternary case, the simplex [4,2,3] 3 code is the shortest minimal code of dimension 2 and we use a [9,3,5] 3 code with generator matrix   1 0 0 1 2 0 0 2 2 0 1 0 0 0 1 2 1 2 0 0 1 1 1 We give here the generator matrix of the [15,6,6] 2 code obtained by the concatenation of the extended Reed-Solomon [5,3,3] 4 with the simplex [3, 2, 2] 2 code:…”

Higgledy-piggledy sets in projective spaces of small dimension

“…Let S be a point set of PG(N, q). The authors of [7,Corollary 13] proved that Γ P,H (4, q) has a resolving set of size 12q if q > 36086 is no power of 2 or 3 (as a corollary of Theorem 1.10(2.)). We can slightly extend and improve this result, as well as translate the existing result concerning higgledy-piggledy line sets of PG (5, q) to this graph-theoretical context.…”