Coding for Errors and Erasures in Random Network Coding

The problem of error-control in random linear network coding is considered. A ``noncoherent'' or ``channel oblivious'' model is assumed where neither transmitter nor receiver is assumed to have knowledge of the channel transfer characteristic. Motivated by the property that linear network coding is vector-space preserving, information transmission is modelled as the injection into the network of a basis for a vector space $V$ and the collection by the receiver of a basis for a vector space $U$. A metric on the projective geometry associated with the packet space is introduced, and it is shown that a minimum distance decoder for this metric achieves correct decoding if the dimension of the space $V \cap U$ is sufficiently large. If the dimension of each codeword is restricted to a fixed integer, the code forms a subset of a finite-field Grassmannian, or, equivalently, a subset of the vertices of the corresponding Grassmann graph. Sphere-packing and sphere-covering bounds as well as a generalization of the Singleton bound are provided for such codes. Finally, a Reed-Solomon-like code construction, related to Gabidulin's construction of maximum rank-distance codes, is described and a Sudan-style ``list-1'' minimum distance decoding algorithm is provided.Comment: This revised paper contains some minor changes and clarification

Precise Algorithm to Generate Random Sequential Addition of Hard Hyperspheres at Saturation

Random sequential addition (RSA) time-dependent packing process, in which congruent hard hyperspheres are randomly and sequentially placed into a system without interparticle overlap, is a useful packing model to study disorder in high dimensions. Of particular interest is the infinite-time {\it saturation} limit in which the available space for another sphere tends to zero. However, the associated saturation density has been determined in all previous investigations by extrapolating the density results for near-saturation configurations to the saturation limit, which necessarily introduces numerical uncertainties. We have refined an algorithm devised by us [S. Torquato, O. Uche, and F.~H. Stillinger, Phys. Rev. E {\bf 74}, 061308 (2006)] to generate RSA packings of identical hyperspheres. The improved algorithm produce such packings that are guaranteed to contain no available space using finite computational time with heretofore unattained precision and across the widest range of dimensions ($2 \le d \le 8$). We have also calculated the packing and covering densities, pair correlation function $g_2(r)$ and structure factor $S(k)$ of the saturated RSA configurations. As the space dimension increases, we find that pair correlations markedly diminish, consistent with a recently proposed "decorrelation" principle, and the degree of "hyperuniformity" (suppression of infinite-wavelength density fluctuations) increases. We have also calculated the void exclusion probability in order to compute the so-called quantizer error of the RSA packings, which is related to the second moment of inertia of the average Voronoi cell. Our algorithm is easily generalizable to generate saturated RSA packings of nonspherical particles

Periodicity and Circle Packing in the Hyperbolic Plane

We prove that given a fixed radius $r$, the set of isometry-invariant probability measures supported on ``periodic'' radius $r$-circle packings of the hyperbolic plane is dense in the space of all isometry-invariant probability measures on the space of radius $r$-circle packings. By a periodic packing, we mean one with cofinite symmetry group. As a corollary, we prove the maximum density achieved by isometry-invariant probability measures on a space of radius $r$-packings of the hyperbolic plane is the supremum of densities of periodic packings. We also show that the maximum density function varies continuously with radius.Comment: 25 page

Characterization of Maximally Random Jammed Sphere Packings: II. Correlation Functions and Density Fluctuations

In the first paper of this series, we introduced Voronoi correlation functions to characterize the structure of maximally random jammed (MRJ) sphere packings across length scales. In the present paper, we determine a variety of correlation functions that can be rigorously related to effective physical properties of MRJ sphere packings and compare them to the corresponding statistical descriptors for overlapping spheres and equilibrium hard-sphere systems. Such structural descriptors arise in rigorous bounds and formulas for effective transport properties, diffusion and reactions constants, elastic moduli, and electromagnetic characteristics. First, we calculate the two-point, surface-void, and surface-surface correlation functions, for which we derive explicit analytical formulas for finite hard-sphere packings. We show analytically how the contacts between spheres in the MRJ packings translate into distinct functional behaviors of these two-point correlation functions that do not arise in the other two models examined here. Then, we show how the spectral density distinguishes the MRJ packings from the other disordered systems in that the spectral density vanishes in the limit of infinite wavelengths. These packings are hyperuniform, which means that density fluctuations on large length scales are anomalously suppressed. Moreover, we study and compute exclusion probabilities and pore size distributions as well as local density fluctuations. We conjecture that for general disordered hard-sphere packings, a central limit theorem holds for the number of points within an spherical observation window. Our analysis links problems of interest in material science, chemistry, physics, and mathematics. In the third paper, we will evaluate bounds and estimates of a host of different physical properties of the MRJ sphere packings based on the structural characteristics analyzed in this paper.Comment: 25 pages, 13 Figures; corrected typos, updated reference

Highly saturated packings and reduced coverings

We introduce and study certain notions which might serve as substitutes for maximum density packings and minimum density coverings. A body is a compact connected set which is the closure of its interior. A packing $\cal P$ with congruent replicas of a body $K$ is $n$-saturated if no $n-1$ members of it can be replaced with $n$ replicas of $K$, and it is completely saturated if it is $n$-saturated for each $n\ge 1$. Similarly, a covering $\cal C$ with congruent replicas of a body $K$ is $n$-reduced if no $n$ members of it can be replaced by $n-1$ replicas of $K$ without uncovering a portion of the space, and it is completely reduced if it is $n$-reduced for each $n\ge 1$. We prove that every body $K$ in $d$-dimensional Euclidean or hyperbolic space admits both an $n$-saturated packing and an $n$-reduced covering with replicas of $K$. Under some assumptions on $K\subset \mathbb{E}^d$ (somewhat weaker than convexity), we prove the existence of completely saturated packings and completely reduced coverings, but in general, the problem of existence of completely saturated packings and completely reduced coverings remains unsolved. Also, we investigate some problems related to the the densities of $n$-saturated packings and $n$-reduced coverings. Among other things, we prove that there exists an upper bound for the density of a $d+2$-reduced covering of $\mathbb{E}^d$ with congruent balls, and we produce some density bounds for the $n$-saturated packings and $n$-reduced coverings of the plane with congruent circles

Notions of denseness

The notion of a completely saturated packing [Fejes Toth, Kuperberg and Kuperberg, Highly saturated packings and reduced coverings, Monats. Math. 125 (1998) 127-145] is a sharper version of maximum density, and the analogous notion of a completely reduced covering is a sharper version of minimum density. We define two related notions: uniformly recurrent and weakly recurrent dense packings, and diffusively dominant packings. Every compact domain in Euclidean space has a uniformly recurrent dense packing. If the domain self-nests, such a packing is limit-equivalent to a completely saturated one. Diffusive dominance is yet sharper than complete saturation and leads to a better understanding of n-saturation.Comment: Published in Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol4/paper9.abs.htm

Intrinsic circle domains

Using quasiconformal mappings, we prove that any Riemann surface of finite connectivity and finite genus is conformally equivalent to an intrinsic circle domain Ω \Omega in a compact Riemann surface S S . This means that each connected component B B of S ∖ Ω S\setminus \Omega is either a point or a closed geometric disc with respect to the complete constant curvature conformal metric of the Riemann surface ( Ω ∪ B ) (\Omega \cup B) . Moreover, the pair ( Ω , S ) (\Omega , S) is unique up to conformal isomorphisms. We give a generalization to countably infinite connectivity. Finally, we show how one can compute numerical approximations to intrinsic circle domains using circle packings and conformal welding.</p

Epitaxial Frustration in Deposited Packings of Rigid Disks and Spheres

We use numerical simulation to investigate and analyze the way that rigid disks and spheres arrange themselves when compressed next to incommensurate substrates. For disks, a movable set is pressed into a jammed state against an ordered fixed line of larger disks, where the diameter ratio of movable to fixed disks is 0.8. The corresponding diameter ratio for the sphere simulations is 0.7, where the fixed substrate has the structure of a (001) plane of a face-centered cubic array. Results obtained for both disks and spheres exhibit various forms of density-reducing packing frustration next to the incommensurate substrate, including some cases displaying disorder that extends far from the substrate. The disk system calculations strongly suggest that the most efficient (highest density) packings involve configurations that are periodic in the lateral direction parallel to the substrate, with substantial geometric disruption only occurring near the substrate. Some evidence has also emerged suggesting that for the sphere systems a corresponding structure doubly periodic in the lateral directions would yield the highest packing density; however all of the sphere simulations completed thus far produced some residual "bulk" disorder not obviously resulting from substrate mismatch. In view of the fact that the cases studied here represent only a small subset of all that eventually deserve attention, we end with discussion of the directions in which first extensions of the present simulations might profitably be pursued.Comment: 28 pages, 14 figures; typos fixed; a sentence added to 4th paragraph of sect 5 in responce to a referee's comment