192 research outputs found
New results on torus cube packings and tilings
We consider sequential random packing of integral translate of cubes
into the torus . Two special cases are of special
interest:
(i) The case which corresponds to a discrete case of tilings
(considered in \cite{cubetiling,book})
(ii) The case corresponds to a case of continuous tilings
(considered in \cite{combincubepack,book})
Both cases correspond to some special combinatorial structure and we describe
here new developments.Comment: 5 pages, conference pape
A finite subdivision rule for the n-dimensional torus
Cannon, Floyd, and Parry have studied subdivisions of the 2-sphere
extensively, especially those corresponding to 3-manifolds, in an attempt to
prove Cannon's conjecture. There has been a recent interest in generalizing
some of their tools, such as extremal length, to higher dimensions. We define
finite subdivision rules of dimension n, and find an n-1-dimensional finite
subdivision rule for the n-dimensional torus, using a well-known simplicial
decomposition of the hypercube. We hope to expand on this and find finite
subdivision rules for many higher-dimensional manifolds, including hyperbolic
n-manifolds.Comment: Accepted by Geometriae Dedicata; ublished version available onlin
Interleaving schemes for multidimensional cluster errors
We present two-dimensional and three-dimensional interleaving techniques for correcting two- and three-dimensional bursts (or clusters) of errors, where a cluster of errors is characterized by its area or volume. Correction of multidimensional error clusters is required in holographic storage, an emerging application of considerable importance. Our main contribution is the construction of efficient two-dimensional and three-dimensional interleaving schemes. The proposed schemes are based on t-interleaved arrays of integers, defined by the property that every connected component of area or volume t consists of distinct integers. In the two-dimensional case, our constructions are optimal: they have the lowest possible interleaving degree. That is, the resulting t-interleaved arrays contain the smallest possible number of distinct integers, hence minimizing the number of codewords required in an interleaving scheme. In general, we observe that the interleaving problem can be interpreted as a graph-coloring problem, and introduce the useful special class of lattice interleavers. We employ a result of Minkowski, dating back to 1904, to establish both upper and lower bounds on the interleaving degree of lattice interleavers in three dimensions. For the case tâĄ0 mod 6, the upper and lower bounds coincide, and the Minkowski lattice directly yields an optimal lattice interleaver. For tâ 0 mod 6, we construct efficient lattice interleavers using approximations of the Minkowski lattice
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