Sphere packing and shannons theorem in the rst section we discuss the basics of block coding on the mary symmetric channel. The spherepacking argument only yields the maximum number of code. The problem of correcting for errors on noisy communication channels like this is exactly the spherepacking problem, said henry cohn, a mathematician at microsoft research new england in cambridge, mass. Block codes, error exponent, list decoding, spherepacking bound, turbolike codes. The following theorem shows that when packing unit spheres in a large box the spheres occupy at most about 0.
Carl friedrich gauss proved that the highest average density that is, the greatest fraction of space occupied by spheres that can be achieved by a lattice packing is. But, for information theory newbies, the socalled sphere packing argument is welcomed due to its intuitiveness. Information theory is the study of achievable bounds for communication and is largely probabilistic and analytic in nature. Capacity of the gaussian arbitrarily varying channel. Some ternary and quaternary codes and associated sphere. Advances in random matrix theory, zeta functions, and. Abstract high performance concrete hpc has became more popular in recent years. The spherepacking arrangement is called the coding, and for more than. Shannons application of highdimensional sphere packings. In the second section we see how the geometry of the codespace can be used to make coding judgements. Information theory is therefore a theory about biology, and shannon was a biologist. In one dimension it is packing line segments into a linear universe.
The same packing density can also be achieved by alternate stackings of the same. However, sphere packing problems can be generalised to consider unequal spheres, spaces of other dimensions or to noneuclidean spaces such as hyperbolic space. Information theory shannons application of highdimensional sphere packings. Gallagers bound is derived in order to show that these sphere packing bounds are. Here, we describe its appearance in number theory, in particular in the theory of zeta functions. Martinez are with the department of information and communication technologies, universitat pompeu. Sequence logos and sequence walkers are explained in a gentle introduction to information theory pdf. His theory was set up for sphere shaped particles and was based on the assumption that the small particles fill out the cavities between the big particles without disturbing the packing of the big particles. The shape quantizer is constructed from wrapped spherical codes that map a sphere packing in 1 onto a sphere in, and the gain codebook is a. We consider packings with m spheres in the unit cell mperiodic packings. Our goal here is to improve the linear bound by a large constant, and also to. Index termsjoint sourcechannel coding, reliability function. Classical random matrix theory is concerned with statistical questions about the eigenvalues of large matrices in various ensembles.
Between sphere packing and sphere covering caroline uhler ist austria joint work with mabel iglesiasham, michael kerber, stephen j. Ld convergence, quotients, and right convergence with christian borgs, jennifer. Ray liu, fellow, ieee abstractorthogonal designs have received considerable attention in the development of ef. Review of particle packing theories used for concrete mix proportioning. An improved spherepacking bound for finitelength codes over. The standard ensembles are the circular unitary, orthogonal, and symplectic ensembles cue, coe, and cse. Sphere packing is the problem of arranging nonoverlapping spheres within some space, with the goal of maximizing the combined volume of the spheres. Suppose there are n,, centers x with ii x ii m in a sphere packing a. Almost 4 centuries ago, kepler studied the densities of sphere packings and made his famous conjecture. Pdf, dvi and plain tex files of papers and preprints by r.
Ray liu, fellow, ieee abstractorthogonal designs have received considerable at tention in the development of ef. By erica klarreich in a pair of papers posted online this month, a ukrainian mathematician has solved two highdimensional versions of the centuriesold sphere packing problem. Claude shannon 19162001 developed information theory. Thomas harriot 15601621 mathematical assistant to sir walter raleigh. The following theorem shows that when packing unit spheres in a large box the spheres occupy at. In geometry, a sphere packing is an arrangement of nonoverlapping spheres within a containing space. Universally optimal distribution of points on spheres with abhinav kumar journal of the american mathematical society 20 2007, 99148, arxiv. A brief and true report of the new found land of virginia 1588 first to study the sphere packing problem. Sphere packing is a rst step towards these more complex problems. This problem has its roots in geometry, number theory and information theory and it is part of hilberts 18th problem. Bounds on packings of spheres in the grassmann manifold. This can be directly verified from shannons channel coding theorem with mutual information computation.
Whats the densest sphere packing in a million dimensions. Biological information theory and the theory of molecular. We say that provides a k fold packing for caps of radius. The density is the fraction of space covered by the balls, and the problem is to nd the maximal possible density. Gaussian channel gaussian channel gaussian channel capacity dr. A typical sphere packing problem is to find an arrangement in which the spheres fill as much of the space as possible. Construction of spherical codes from sphere packings the construction is very simple. The sphere packing problem asks for the densest packing of unit balls in e d. Similarly, balls result yields a sphere packing of density at least 2n 12 nin every dimension. Mona lisa in awgn mona lisa 200 400 600 100 200 300 400 500 600 700 800 900 1100. Yao xie, ece587, information theory, duke university. Handwritten notes pdf study material for all engineering mathematics students. Coding theory then attempts to realize the promise of these bounds by models which are constructed through mainly algebraic means. Furnas considered the ideal packing of a mixture of two materials.
Logos are only the beginning, however, as the information theory measure used to compute them gives results in bits. Locally optimal 2periodic sphere packings springerlink. Lecture notes in information theory volume ii by poning chenyand fady alajajiz ydepartment of electrical engineering institute of communication engineering national chiao tung university 1001, ta hsueh road hsin chu, taiwan 30056 republic of china email. But why would a binding site have some number of bits. In the course of centuries, many exciting results have been obtained, ingenious methods created, related. But the optimal codes for these forms of transmission correspond to the densest packing of spheres in dimensions beyond the three of everyday experience, and higherdimensional problems have proved formidable. Tom schneider is best known for inventing sequence logos, a computer graphic depicting patterns in dna, rna or protein that is now widely used by molecular biologists. I, january 1991 capacity of the gaussian arbitrarily varying channel imre csiszhr and prakash narayan, member, zeee abstract the gaussian arbitrarily varying channel with input con straint r and state constraint 2 admits input sequences x xl, x, of real numbers with cxf 5 nt and state sequences s 5.
In the classical case, the spheres are all of the same sizes, and the space in question is threedimensional space e. The most important question in this area is keplers problem. Multisized sphere packing louisiana tech university. Shannons channel capacity theorem only applies to living organisms and their products, such as communications channels and molecular machines that make choices from several possibilities.
Shannon was primarily interested in the information theory. Sphere packings is one of the most attractive and challenging subjects in mathematics. This connection is perhaps best exempli ed by a striking similarity between two fundamental inequalities in the elds. The ieee information theory society is an international organization whose purpose is to connect people interested in processing, transmission, storage, and use of information, as well as theoretical and applied aspects of coding, communications, and communications networks. Wright algebraic statistics meeting iit chicago may 19, 2014 caroline uhler ist austria packing spheres algebraic statistics, may 2014 1 16. Basic research of packing theory was started by furnas 6. In two dimensions, the equivalent problem is packing circles on a plane.
Multiple packing of spherical caps mathematika cambridge core. Sphere packing and kissing numbers problems of arranging balls densely arise in many situations, particularly in coding theory the balls are formed by the sets of inputs that the errorcorrection would map into a single codeword. Sphere packing solved in higher dimensions a ukrainian mathematician has solved the centuriesold spherepacking problem in dimensions eight and 24. Spherepacking problems underlie almost all digital communications and storage, from cell phones to cds to the internet. The sphere packing problem asks how to arrange congruent balls as densely as possible without overlap between their interiors.
The paper was initiated by 1, where typical properties of random lattices and random packings of a euclidean space were studied. The spheres considered are usually all of identical size, and the space is usually threedimensional euclidean space. Kschischang, member, ieee, and subbarayan pasupathy, fellow, ieee abstracttables are presented of good ternary and quater. The problem of correcting for errors on noisy communication channels like this is exactly the spherepacking problem, said henry cohn, a mathematician at microsoft research new england in cambridge, mass spherepacking problems underlie almost all digital communications and storage, from cell phones to cds to the internet. This problem plays an important role in geometry, number theory, and information theory. Henry cohn iap math lecture series january 16, 2015. Yao xie, ece587, information theory, duke university 9. We use other simpler and more precise ways to obtain estimates on parameters that characterize random packings and consider the possibility of extending the results to lpackings. Mathematical progress on spherepacking optimization problems.
Lecture notes in information theory volume ii by poning chenyand fady alajajiz. Sphere packing, modular forms, fourier analysis ams subject classi cation. Encoding of linear codes, decoding of linear codes, singleton bound and maximum distance separable codes, spherepacking bound and perfect codes, gilbert. Gaussian source coding with spherical codes information. The sphere packing argument only yields the maximum number of code.
Information theory is the study of achievable bounds for communication and is largely probabilistic and analytic in. This far surpasses the density of any \explicitly known sphere packing. Review of particle packing theories used for concrete mix. Sphere packing is a rst step towards these more complex. Multisized sphere packing shuji yamada1 jinko kanno2 miki miyauchi3 1department of computer science kyoto sangyo university, japan 2mathematics and statistics program louisiana tech university, u. Concepts in geometry often have parallels in information theory. A brief report on recent work on the spherepacking problem. Kschischang, member, ieee, and subbarayan pasupathy, fellow, ieee abstracttables are presented of good ternary and quater nary codes and they are used in the construction of dense sphere packings. We study probabilistic characteristics of random packings of a euclidean space. Sphere packing solved in higher dimensions quanta magazine. The sphere packing problem in dimension 8 maryna s.
The sphere packing problem is the threedimensional version of a class of ball packing problems in arbitrary dimensions. This leads to the third section where we present some information theory and shannons basic channel coding theorem. The packing density of large lattice packings of spheres in euclidean e d measured by the parametric density depends on the parameter and on the shape of the convex hull p of the sphere centers. In geometry, closepacking of equal spheres is a dense arrangement of congruent spheres in an infinite, regular arrangement or lattice. Let k be a positive integer and suppose that no point of the sphere is an inner point of more than k caps. Gaussian source coding with spherical codes jon hamkins, member, ieee, and kenneth zeger, fellow, ieee abstract a fixedrate shapegain quantizer for the memoryless gaussian source is proposed. Viazovska april 5, 2017 in this paper we prove that no packing of unit balls in euclidean space r8 has density greater than that of the e 8lattice packing. Mar 30, 2016 mathematicians have been studying sphere packings since at least 1611, when johannes kepler conjectured that the densest way to pack together equalsized spheres in space is the familiar pyramidal piling of oranges seen in grocery stores. Biologist the founder of information theory used biology to formulate the channel capacity by thomas d. Pdf a sphere packing bound spb with a prefactor that is polynomial in the block length n is. Mathematical progress on spherepacking optimization. The various performance attributes of hpc such as strength. For the case \m1\ lattice packings, voronoi proved there are finitely many inequivalent local optima and presented an algorithm to enumerate them, and this computation has been implemented in up to \d8\ dimensions.
21 1048 259 1549 290 1258 997 120 1245 260 4 1221 368 459 799 886 194 1336 1645 1142 229 953 1454 884 1212 874 772 1222 133 459 1201 15