9 research outputs found

    Lines in higgledy-piggledy arrangement

    Get PDF

    Resolving sets for higher dimensional projective spaces

    Get PDF
    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 [9,4]2[9,4]_{2}-codes are hyperbolic quadrics

    Full text link
    An [n,k]q[n,k]_{q}-linear code is a kk-dimensional subspace of Fqn\mathbb{F}_{q}^{n}. A codeword cc is minimal if its support does not contain the support of any codeword c′≠λcc'\neq\lambda c, λ∈Fq∗\lambda\in \mathbb{F}^*_{q}. A linear code is minimal if all its codewords are minimal. [n,k]q[n,k]_{q}-minimal linear codes are in bijection with strong blocking sets of size nn in PG(k−1,q)PG(k-1,q) and a lower bound for the size of strong blocking set is given by (k−1)(q+1)≤n(k-1)(q+1)\leq n. In this note we show that all strong blocking set of length 9 in PG(3,2)PG(3,2) are the hyperbolic quadrics Q+(3,2)Q^{+}(3,2)

    Higgledy-piggledy sets in projective spaces of small dimension

    Full text link
    This work focuses on higgledy-piggledy sets of kk-subspaces in PG(N,q)\text{PG}(N,q), i.e. sets of projective subspaces that are 'well-spread-out'. More precisely, the set of intersection points of these kk-subspaces with any (N−k)(N-k)-subspace κ\kappa of PG(N,q)\text{PG}(N,q) spans κ\kappa itself. We highlight three methods to construct small higgledy-piggledy sets of kk-subspaces and discuss, for k∈{1,N−2}k\in\{1,N-2\}, 'optimal' sets that cover the smallest possible number of points. Furthermore, we investigate small non-trivial higgledy-piggledy sets in PG(N,q)\text{PG}(N,q), N⩽5N\leqslant5. Our main result is the existence of six lines of PG(4,q)\text{PG}(4,q) in higgledy-piggledy arrangement, two of which intersect. Exploiting the construction methods mentioned above, we also show the existence of six planes of PG(4,q)\text{PG}(4,q) in higgledy-piggledy arrangement, two of which maximally intersect, as well as the existence of two higgledy-piggledy sets in PG(5,q)\text{PG}(5,q) 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

    Full text link
    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

    Full text link
    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 (k−1)(k-1)-dimensional projective space over Fq\mathbb{F}_q that have size O(qk)O( q k ). Since strong blocking sets have recently been shown to be equivalent to minimal linear codes, our construction gives the first explicit construction of Fq\mathbb{F}_q-linear minimal codes of length nn and dimension kk, for every prime power qq, for which n=O(qk)n = O (q k). This solves one of the main open problems on minimal codes.Comment: 20 page

    On subspace designs

    Full text link
    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
    corecore