The sphere packing problem asks for the densest packing of unit balls in e d. 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. 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. Multisized sphere packing louisiana tech university.
A brief and true report of the new found land of virginia 1588 first to study the sphere packing problem. This can be directly verified from shannons channel coding theorem with mutual information computation. But why would a binding site have some number of bits. This problem has its roots in geometry, number theory and information theory and it is part of hilberts 18th problem. A typical sphere packing problem is to find an arrangement in which the spheres fill as much of the space as possible. The standard ensembles are the circular unitary, orthogonal, and symplectic ensembles cue, coe, and cse.
Information theory is the study of achievable bounds for communication and is largely probabilistic and analytic in nature. Sphere packings is one of the most attractive and challenging subjects in mathematics. 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. Whats the densest sphere packing in a million dimensions. Henry cohn iap math lecture series january 16, 2015. In geometry, a sphere packing is an arrangement of nonoverlapping spheres within a containing space. Information theory shannons application of highdimensional sphere packings. Gallagers bound is derived in order to show that these sphere packing bounds are. 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. Sphere packing and shannons theorem in the rst section we discuss the basics of block coding on the mary symmetric channel. We consider packings with m spheres in the unit cell mperiodic packings. The density is the fraction of space covered by the balls, and the problem is to nd the maximal possible density.
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. Biological information theory and the theory of molecular. Spherepacking problems underlie almost all digital communications and storage, from cell phones to cds to the internet. 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. Concepts in geometry often have parallels in information theory. 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.
Ray liu, fellow, ieee abstractorthogonal designs have received considerable attention in the development of ef. Yao xie, ece587, information theory, duke university. This leads to the third section where we present some information theory and shannons basic channel coding theorem. Logos are only the beginning, however, as the information theory measure used to compute them gives results in bits. Let k be a positive integer and suppose that no point of the sphere is an inner point of more than k caps. Advances in random matrix theory, zeta functions, and. 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. Block codes, error exponent, list decoding, spherepacking bound, turbolike codes. Yao xie, ece587, information theory, duke university 9.
Claude shannon 19162001 developed information theory. Bounds on packings of spheres in the grassmann manifold. 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. Kschischang, member, ieee, and subbarayan pasupathy, fellow, ieee abstracttables are presented of good ternary and quater. This problem plays an important role in geometry, number theory, and information theory.
Abstract high performance concrete hpc has became more popular in recent years. In one dimension it is packing line segments into a linear universe. Thomas harriot 15601621 mathematical assistant to sir walter raleigh. Shannon was primarily interested in the information theory. The sphere packing argument only yields the maximum number of code. 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. We say that provides a k fold packing for caps of radius. Capacity of the gaussian arbitrarily varying channel.
In the second section we see how the geometry of the codespace can be used to make coding judgements. 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. Gaussian source coding with spherical codes information. In two dimensions, the equivalent problem is packing circles on a plane. In the course of centuries, many exciting results have been obtained, ingenious methods created, related. Wright algebraic statistics meeting iit chicago may 19, 2014 caroline uhler ist austria packing spheres algebraic statistics, may 2014 1 16. The same packing density can also be achieved by alternate stackings of the same. Sphere packing, modular forms, fourier analysis ams subject classi cation. 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.
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. The most important question in this area is keplers problem. Sphere packing is a rst step towards these more complex. Multiple packing of spherical caps mathematika cambridge core. An improved spherepacking bound for finitelength codes over. Martinez are with the department of information and communication technologies, universitat pompeu.
Sphere packing is the problem of arranging nonoverlapping spheres within some space, with the goal of maximizing the combined volume of the spheres. Our goal here is to improve the linear bound by a large constant, and also to. The sphere packing problem in dimension 8 maryna s. The paper was initiated by 1, where typical properties of random lattices and random packings of a euclidean space were studied. Here, we describe its appearance in number theory, in particular in the theory of zeta functions. Gaussian channel gaussian channel gaussian channel capacity dr. Between sphere packing and sphere covering caroline uhler ist austria joint work with mabel iglesiasham, michael kerber, stephen j. Furnas considered the ideal packing of a mixture of two materials. Sphere packing solved in higher dimensions quanta magazine. Universally optimal distribution of points on spheres with abhinav kumar journal of the american mathematical society 20 2007, 99148, arxiv. Mona lisa in awgn mona lisa 200 400 600 100 200 300 400 500 600 700 800 900 1100. Mathematical progress on spherepacking optimization. The spheres considered are usually all of identical size, and the space is usually threedimensional euclidean space. Basic research of packing theory was started by furnas 6.
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. Information theory is therefore a theory about biology, and shannon was a biologist. Review of particle packing theories used for concrete mix proportioning. Lecture notes in information theory volume ii by poning chenyand fady alajajiz. The following theorem shows that when packing unit spheres in a large box the spheres occupy at. The following theorem shows that when packing unit spheres in a large box the spheres occupy at most about 0. This connection is perhaps best exempli ed by a striking similarity between two fundamental inequalities in the elds. This far surpasses the density of any \explicitly known sphere packing. Shannons application of highdimensional sphere packings. Sphere packing solved in higher dimensions a ukrainian mathematician has solved the centuriesold spherepacking problem in dimensions eight and 24. Encoding of linear codes, decoding of linear codes, singleton bound and maximum distance separable codes, spherepacking bound and perfect codes, gilbert. In geometry, closepacking of equal spheres is a dense arrangement of congruent spheres in an infinite, regular arrangement or lattice. 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.
Sequence logos and sequence walkers are explained in a gentle introduction to information theory pdf. The spherepacking argument only yields the maximum number of code. Coding theory then attempts to realize the promise of these bounds by models which are constructed through mainly algebraic means. A brief report on recent work on the spherepacking problem. The sphere packing problem asks how to arrange congruent balls as densely as possible without overlap between their interiors. Some ternary and quaternary codes and associated sphere.
Ray liu, fellow, ieee abstractorthogonal designs have received considerable at tention in the development of ef. But, for information theory newbies, the socalled sphere packing argument is welcomed due to its intuitiveness. Biologist the founder of information theory used biology to formulate the channel capacity by thomas d. The sphere packing problem is the threedimensional version of a class of ball packing problems in arbitrary dimensions.
Pdf, dvi and plain tex files of papers and preprints by r. Ld convergence, quotients, and right convergence with christian borgs, jennifer. Handwritten notes pdf study material for all engineering mathematics students. But, for information theory newbies, the socalled spherepacking argument is welcomed due to its intuitiveness. The spherepacking arrangement is called the coding, and for more than. 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.
Review of particle packing theories used for concrete mix. However, sphere packing problems can be generalised to consider unequal spheres, spaces of other dimensions or to noneuclidean spaces such as hyperbolic space. Classical random matrix theory is concerned with statistical questions about the eigenvalues of large matrices in various ensembles. Almost 4 centuries ago, kepler studied the densities of sphere packings and made his famous conjecture. 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. Suppose there are n,, centers x with ii x ii m in a sphere packing a. 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. Index termsjoint sourcechannel coding, reliability function. Similarly, balls result yields a sphere packing of density at least 2n 12 nin every dimension.
Information theory is the study of achievable bounds for communication and is largely probabilistic and analytic in. Locally optimal 2periodic sphere packings springerlink. The various performance attributes of hpc such as strength. Sphere packing is a rst step towards these more complex problems. 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. Construction of spherical codes from sphere packings the construction is very simple. 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. We study probabilistic characteristics of random packings of a euclidean space. Mathematical progress on spherepacking optimization problems.
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