He can be reached at yahoo dot com with email address starting with fyc and followed by his last name. Why you should care about highdimensional sphere packing. Sloane, new bounds on the number of unit spheres that can touch a unit sphere in n dimensions, j. These include the use of finite groups and sphere packings in highdimensional spaces for the design of error correcting codes. In this way we construct the densest packings known in all dimensions n.
Error correcting codes are used in several constructions for packings of equal spheres in dimensional euclidean spaces en. Bounds on errorcorrecting codes can be translated into sphere packing, to place rigorous upper bounds on. Sloane error correcting codes are used in several constructions for packings of equal spheres in dimensional euclidean spacen. Sphere packing and spherical codes 329 so we may take the radius of the spheres to be p l2.
Sphere packing and errorcorrecting codes request pdf. Sphere packings, crystals, and errorcorrecting codes what is the. Sphere packing and errorcorrecting codes adex theory. Thess e e include a systematic derivation of many of the best sphere packings known, and construction of. Errorcorrecting codes are used to construct dense sphere packings in n dimensional euclidean space rn, and other packings are obtained by taking. A new class of spherical codes called wrapped spher ical codes is constructed by wrapping any sphere packing in euclidean space onto a finite subset of the unit sphere in one higher dimension. A necessary condition for the existence of a perfect binary code which can correct more than one error, is the existence of three or more.
Sphere packing and errorcorrecting codes springerlink. The 7, 4 hamming code is a perfect 1error correcting code, as we shall see in section 4. The sphere packing problem in dimension 8 maryna s. In exceptional cases linear programming bounds are optimal 5. Just as a binary error correcting code 36 is a subset of the vertices of an ndimensional cube, so a spherical code 16, 21 is a subset of the points an ndimensional. Errorcorrecting codes are used in several constructions for packings of equal spheres in dimensional euclidean spaces. From errorcorrecting codes through sphere packings to simple. From errorcorrecting codes through sphere packings to. Error correcting codes are used to construct dense sphere packings in ndimensional euclidean space r n, and other packings are obtained by taking crosssections or building up by layers. Errorcorrecting codes are used to construct dense sphere packings in ndimensional euclidean space r n, and other packings are obtained by taking. Improving the spherepacking bound for binary codes over memory less symmetric channels conference paper pdf available november 2009 with 74 reads how we measure reads. It is a laminated lattice but can also be constructed from a 24,12,8code. Errorcorrecting codes are used to construct dense sphere packings in n dimensional euclidean space r n, and other packings are obtained by taking. These include a systematic derivation of many of the best sphere.
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