321 research outputs found

    Parity of Sets of Mutually Orthogonal Latin Squares

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    Every Latin square has three attributes that can be even or odd, but any two of these attributes determines the third. Hence the parity of a Latin square has an information content of 2 bits. We extend the definition of parity from Latin squares to sets of mutually orthogonal Latin squares (MOLS) and the corresponding orthogonal arrays (OA). Suppose the parity of an OA(k,n)\mathrm{OA}(k,n) has an information content of dim(k,n)\dim(k,n) bits. We show that dim(k,n)(k2)1\dim(k,n) \leq {k \choose 2}-1. For the case corresponding to projective planes we prove a tighter bound, namely dim(n+1,n)(n2)\dim(n+1,n) \leq {n \choose 2} when nn is odd and dim(n+1,n)(n2)1\dim(n+1,n) \leq {n \choose 2}-1 when nn is even. Using the existence of MOLS with subMOLS, we prove that if dim(k,n)=(k2)1\dim(k,n)={k \choose 2}-1 then dim(k,N)=(k2)1\dim(k,N) = {k \choose 2}-1 for all sufficiently large NN. Let the ensemble of an OA\mathrm{OA} be the set of Latin squares derived by interpreting any three columns of the OA as a Latin square. We demonstrate many restrictions on the number of Latin squares of each parity that the ensemble of an OA(k,n)\mathrm{OA}(k,n) can contain. These restrictions depend on nmod4n\mod4 and give some insight as to why it is harder to build projective planes of order n2mod4n \not= 2\mod4 than for n2mod4n \not= 2\mod4. For example, we prove that when n2mod4n \not= 2\mod 4 it is impossible to build an OA(n+1,n)\mathrm{OA}(n+1,n) for which all Latin squares in the ensemble are isotopic (equivalent to each other up to permutation of the rows, columns and symbols)

    Fractional repetition codes with flexible repair from combinatorial designs

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    Fractional repetition (FR) codes are a class of regenerating codes for distributed storage systems with an exact (table-based) repair process that is also uncoded, i.e., upon failure, a node is regenerated by simply downloading packets from the surviving nodes. In our work, we present constructions of FR codes based on Steiner systems and resolvable combinatorial designs such as affine geometries, Hadamard designs and mutually orthogonal Latin squares. The failure resilience of our codes can be varied in a simple manner. We construct codes with normalized repair bandwidth (β\beta) strictly larger than one; these cannot be obtained trivially from codes with β=1\beta = 1. Furthermore, we present the Kronecker product technique for generating new codes from existing ones and elaborate on their properties. FR codes with locality are those where the repair degree is smaller than the number of nodes contacted for reconstructing the stored file. For these codes we establish a tradeoff between the local repair property and failure resilience and construct codes that meet this tradeoff. Much of prior work only provided lower bounds on the FR code rate. In our work, for most of our constructions we determine the code rate for certain parameter ranges.Comment: 27 pages in IEEE two-column format. IEEE Transactions on Information Theory (to appear

    Diagonal groups and arcs over groups

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    Partially supported by Simons Foundation Collaboration Grant 359872 and by Fundacaopara a Ciencia e a Tecnologia (Portuguese Foundation for Science and Technology) grant PTDC/MAT-PUR/31174/2017. Australian Research Council Discovery Grant DP160102323.In an earlier paper by three of the present authors and Csaba Schneider, it was shown that, for m≥2, a set of m+1 partitions of a set Ω, any m of which are the minimal non-trivial elements of a Cartesian lattice, either form a Latin square (if m=2), or generate a join-semilattice of dimension m associated with a diagonal group over a base group G. In this paper we investigate what happens if we have m+r partitions with r≥2, any m of which are minimal elements of a Cartesian lattice. If m=2, this is just a set of mutually orthogonal Latin squares. We consider the case where all these squares are isotopic to Cayley tables of groups, and give an example to show the groups need not be all isomorphic. For m>2, things are more restricted. Any m+1 of the partitions generate a join-semilattice admitting a diagonal group over a group G. It may be that the groups are all isomorphic, though we cannot prove this. Under an extra hypothesis, we show that G must be abelian and must have three fixed-point-free automorphisms whose product is the identity. (We describe explicitly all abelian groups having such automorphisms.) Under this hypothesis, the structure gives an orthogonal array, and conversely in some cases. If the group is cyclic of prime order p, then the structure corresponds exactly to an arc of cardinality m+r in the (m−1)-dimensional projective space over the field with p elements, so all known results about arcs are applicable. More generally, arcs over a finite field of order q give examples where G is the elementary abelian group of order q. These examples can be lifted to non-elementary abelian groups using p-adic techniques.Publisher PDFPeer reviewe

    A Survey of Finite Algebraic Geometrical Structures Underlying Mutually Unbiased Quantum Measurements

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    The basic methods of constructing the sets of mutually unbiased bases in the Hilbert space of an arbitrary finite dimension are discussed and an emerging link between them is outlined. It is shown that these methods employ a wide range of important mathematical concepts like, e.g., Fourier transforms, Galois fields and rings, finite and related projective geometries, and entanglement, to mention a few. Some applications of the theory to quantum information tasks are also mentioned.Comment: 20 pages, 1 figure to appear in Foundations of Physics, Nov. 2006 two more references adde
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