519 research outputs found

    Higgledy-piggledy sets in projective spaces of small dimension

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    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 (Nk)(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,N2}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), N5N\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

    Lines in higgledy-piggledy arrangement

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    Higgledy-piggledy subspaces and uniform subspace designs

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    In this article, we investigate collections of `well-spread-out' projective (and linear) subspaces. Projective kk-subspaces in PG(d,F)\mathsf{PG}(d,\mathbb{F}) are in `higgledy-piggledy arrangement' if they meet each projective subspace of co-dimension kk in a generator set of points. We prove that the set H\mathcal{H} of higgledy-piggledy kk-subspaces has to contain more than minF,i=0kdk+ii+1\min{|\mathbb{F}|,\sum_{i=0}^k\lfloor\frac{d-k+i}{i+1}\rfloor} elements. We also prove that H\mathcal{H} has to contain more than (k+1)(dk)(k+1)\cdot(d-k) elements if the field F\mathbb{F} is algebraically closed. An rr-uniform weak (s,A)(s,A) subspace design is a set of linear subspaces H1,..,HNFmH_1,..,H_N\le\mathbb{F}^m each of rank rr such that each linear subspace WFmW\le\mathbb{F}^m of rank ss meets at most AA among them. This subspace design is an rr-uniform strong (s,A)(s,A) subspace design if i=1Nrank(HiW)A\sum_{i=1}^N\mathrm{rank}(H_i\cap W)\le A for WFm\forall W\le\mathbb{F}^m of rank ss. We prove that if m=r+sm=r+s then the dual ({H1,...,HN}\{H_1^\bot,...,H_N^\bot\}) of an rr-uniform weak (strong) subspace design of parameter (s,A)(s,A) is an ss-uniform weak (strong) subspace design of parameter (r,A)(r,A). We show the connection between uniform weak subspace designs and higgledy-piggledy subspaces proving that AminF,i=0r1s+ii+1A\ge\min{|\mathbb{F}|,\sum_{i=0}^{r-1}\lfloor\frac{s+i}{i+1}\rfloor} for rr-uniform weak or strong (s,A)(s,A) subspace designs in Fr+s\mathbb{F}^{r+s}. We show that the rr-uniform strong (s,rs+(r2))(s,r\cdot s+\binom{r}{2}) subspace design constructed by Guruswami and Kopprty (based on multiplicity codes) has parameter A=rsA=r\cdot s if we consider it as a weak subspace design. We give some similar constructions of weak and strong subspace designs (and higgledy-piggledy subspaces) and prove that the lower bound (k+1)(dk)+1(k+1)\cdot(d-k)+1 over algebraically closed field is tight.Comment: 27 pages. Submitted to Designs Codes and Cryptograph

    Higgledy-piggledy subspaces and uniform subspace designs

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    In this article, we investigate collections of ` well-spread-out' projective (and linear) subspaces. Projective k-subspaces in PG(d, F) are in ` higgledy-piggledy arrangement' if they meet each projective subspace of co-dimension k in a generator set of points. We prove that the higgledy-piggledy set H of k-subspaces has to contain more than min {vertical bar F vertical bar, Sigma(k)(i=0) left perpendicular d-k+I/i+1 right perpendicular} elements. We also prove that H has to contain more than (k + 1) . (d -k) elements if the field F is algebraically closed. An r-uniform weak (s, A) subspace design is a set of linear subspaces H-1 ,..., H-N = min { vertical bar F vertical bar, Sigma(r=1)(i=0) left perpendicular s+i/i+1right perpendicular} for r-uniform weak or strong (s, A) subspace designs in Fr+ s. We show that the r-uniform strong (s, r . s + ((r)(2))) subspace design constructed by Guruswami and Kopparty (based on multiplicity codes) has parameter A = r .s if we consider it as a weak subspace design. We give some similar constructions of weak and strong subspace designs (and higgledy-piggledy subspaces) and prove that the lower bound (k + 1) . (d -k) + 1 over algebraically closed field is tight

    Resolving sets for higher dimensional projective spaces

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    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

    Higgledy-piggledy subspaces and uniform subspace designs

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    In this article, we investigate collections of ` well-spread-out' projective (and linear) subspaces. Projective k-subspaces in PG(d, F) are in ` higgledy-piggledy arrangement' if they meet each projective subspace of co-dimension k in a generator set of points. We prove that the higgledy-piggledy set H of k-subspaces has to contain more than min {vertical bar F vertical bar, Sigma(k)(i=0) left perpendicular d-k+I/i+1 right perpendicular} elements. We also prove that H has to contain more than (k + 1) . (d -k) elements if the field F is algebraically closed. An r-uniform weak (s, A) subspace design is a set of linear subspaces H-1 ,..., H-N = min { vertical bar F vertical bar, Sigma(r=1)(i=0) left perpendicular s+i/i+1right perpendicular} for r-uniform weak or strong (s, A) subspace designs in Fr+ s. We show that the r-uniform strong (s, r . s + ((r)(2))) subspace design constructed by Guruswami and Kopparty (based on multiplicity codes) has parameter A = r .s if we consider it as a weak subspace design. We give some similar constructions of weak and strong subspace designs (and higgledy-piggledy subspaces) and prove that the lower bound (k + 1) . (d -k) + 1 over algebraically closed field is tight

    Small Strong Blocking Sets by Concatenation

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    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

    Turner’s paintings of Venice

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