19,613 research outputs found
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 t≡0 mod 6, the upper and lower bounds coincide, and the Minkowski lattice directly yields an optimal lattice interleaver. For t≠0 mod 6, we construct efficient lattice interleavers using approximations of the Minkowski lattice
Problems on q-Analogs in Coding Theory
The interest in -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
Bounding bubbles: the vertex representation of 3d Group Field Theory and the suppression of pseudo-manifolds
Based on recent work on simplicial diffeomorphisms in colored group field
theories, we develop a representation of the colored Boulatov model, in which
the GFT fields depend on variables associated to vertices of the associated
simplicial complex, as opposed to edges. On top of simplifying the action of
diffeomorphisms, the main advantage of this representation is that the GFT
Feynman graphs have a different stranded structure, which allows a direct
identification of subgraphs associated to bubbles, and their evaluation is
simplified drastically. As a first important application of this formulation,
we derive new scaling bounds for the regularized amplitudes, organized in terms
of the genera of the bubbles, and show how the pseudo-manifolds configurations
appearing in the perturbative expansion are suppressed as compared to
manifolds. Moreover, these bounds are proved to be optimal.Comment: 28 pages, 17 figures. Few typos fixed. Minor corrections in figure 6
and theorem
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