476 research outputs found

    Problems on q-Analogs in Coding Theory

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    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

    Chromatic Numbers of Simplicial Manifolds

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    Higher chromatic numbers χs\chi_s of simplicial complexes naturally generalize the chromatic number χ1\chi_1 of a graph. In any fixed dimension dd, the ss-chromatic number χs\chi_s of dd-complexes can become arbitrarily large for s≤⌈d/2⌉s\leq\lceil d/2\rceil [6,18]. In contrast, χd+1=1\chi_{d+1}=1, and only little is known on χs\chi_s for ⌈d/2⌉<s≤d\lceil d/2\rceil<s\leq d. A particular class of dd-complexes are triangulations of dd-manifolds. As a consequence of the Map Color Theorem for surfaces [29], the 2-chromatic number of any fixed surface is finite. However, by combining results from the literature, we will see that χ2\chi_2 for surfaces becomes arbitrarily large with growing genus. The proof for this is via Steiner triple systems and is non-constructive. In particular, up to now, no explicit triangulations of surfaces with high χ2\chi_2 were known. We show that orientable surfaces of genus at least 20 and non-orientable surfaces of genus at least 26 have a 2-chromatic number of at least 4. Via a projective Steiner triple systems, we construct an explicit triangulation of a non-orientable surface of genus 2542 and with face vector f=(127,8001,5334)f=(127,8001,5334) that has 2-chromatic number 5 or 6. We also give orientable examples with 2-chromatic numbers 5 and 6. For 3-dimensional manifolds, an iterated moment curve construction [18] along with embedding results [6] can be used to produce triangulations with arbitrarily large 2-chromatic number, but of tremendous size. Via a topological version of the geometric construction of [18], we obtain a rather small triangulation of the 3-dimensional sphere S3S^3 with face vector f=(167,1579,2824,1412)f=(167,1579,2824,1412) and 2-chromatic number 5.Comment: 22 pages, 11 figures, revised presentatio
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