28,748 research outputs found

    Covering of Subspaces by Subspaces

    Full text link
    Lower and upper bounds on the size of a covering of subspaces in the Grassmann graph \cG_q(n,r) by subspaces from the Grassmann graph \cG_q(n,k), k≥rk \geq r, are discussed. The problem is of interest from four points of view: coding theory, combinatorial designs, qq-analogs, and projective geometry. In particular we examine coverings based on lifted maximum rank distance codes, combined with spreads and a recursive construction. New constructions are given for q=2q=2 with r=2r=2 or r=3r=3. We discuss the density for some of these coverings. Tables for the best known coverings, for q=2q=2 and 5≤n≤105 \leq n \leq 10, are presented. We present some questions concerning possible constructions of new coverings of smaller size.Comment: arXiv admin note: text overlap with arXiv:0805.352

    Covering Numbers in Linear Algebra

    Full text link
    We compute the minimal cardinality of a covering (resp. an irredundant covering) of a vector space over an arbitrary field by proper linear subspaces. Analogues for affine linear subspaces are also given.Comment: 4 page

    Formal concept analysis and structures underlying quantum logics

    Get PDF
    A Hilbert space HH induces a formal context, the Hilbert formal context H‾\overline H, whose associated concept lattice is isomorphic to the lattice of closed subspaces of HH. This set of closed subspaces, denoted C(H)\mathcal C(H), is important in the development of quantum logic and, as an algebraic structure, corresponds to a so-called ``propositional system'', that is, a complete, atomistic, orthomodular lattice which satisfies the covering law. In this paper, we continue with our study of the Chu construction by introducing the Chu correspondences between Hilbert contexts, and showing that the category of Propositional Systems, PropSys, is equivalent to the category of ChuCorsH\text{ChuCors}_{\mathcal H} of Chu correspondences between Hilbert contextsUniversidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

    An extremal problem for integer sparse recovery

    Full text link
    Motivated by the problem of integer sparse recovery we study the following question. Let AA be an m×dm \times d integer matrix whose entries are in absolute value at most kk. How large can be d=d(m,k)d=d(m,k) if all m×mm \times m submatrices of AA are non-degenerate? We obtain new upper and lower bounds on dd and answer a special case of the problem by Brass, Moser and Pach on covering mm-dimensional k×⋯×kk \times \cdots\times k grid by linear subspaces

    Problems on q-Analogs in Coding Theory

    Full text link
    The interest in qq-analogs of codes and designs has been increased in the last few years as a consequence of their new application in error-correction for random network coding. There are many interesting theoretical, algebraic, and combinatorial coding problems concerning these q-analogs which remained unsolved. The first goal of this paper is to make a short summary of the large amount of research which was done in the area mainly in the last few years and to provide most of the relevant references. The second goal of this paper is to present one hundred open questions and problems for future research, whose solution will advance the knowledge in this area. The third goal of this paper is to present and start some directions in solving some of these problems.Comment: arXiv admin note: text overlap with arXiv:0805.3528 by other author
    • …
    corecore