I. INTRODUCTION Particle packing problems, such as how to fill a volume with given solid objects as densely as possible, are among the most ancient and persistent problems in science and mathematics. A packing is a large collection of non-overlapping solid objects (particles) in d -dimensional Euclidean space R $^{d}$. The packing density φ is defined as the fraction of space R d covered by the particles. Dense ordered and random packings of nonoverlapping (hard) particles have been employed to understand the equilibrium and non-equilibrium structure of a variety many-particle systems, including crystals, glasses, heterogeneous materials and granular media [1–4]. Packing problems in dimensions higher than three attract current interest for retrieving stored data transmitted through a noisy channel [5–8]. The packings of congruent hard spheres in R 3 have been intensively studied since despite the simplicity they exhibit rich packing characteristics. It is only recently that the densest packings with φ$_{max}$ = $_{π/}$√$_{18}$ ≈ 0 . 74, realized by the face-centered cubic lattice and its stacking variants, have been proved [9]. In addition, three-dimensional random packings can be prepared both experimentally and numerically with a relatively robust density φ ≈ 0 . 64 [10, 11]. The term random close packing (RCP) [12], widely used to designate the “random” packing with the highest achievable density, is ill-defined since random packings can be obtained as the system becomes more ordered and a definition of randomness has been lacking. A more recent concept that has been suggested to replace RCP is that of the maximally random jammed (MRJ) state [11], corresponding to the most disordered among all jammed (mechanically stable) packings. A jammed packing is one in which the particle positions and orientations are fixed by the impenetrability constraints and boundary conditions [13]. It has been established that the MRJ state for spheres in R 3 has a density of φ ≈ 0 . 637, as obtained by a variety of different order metrics [13, 14]. This density value is consistent with what has traditionally asscoiated with RCP in three dimensions. It has been argued in the granular materials literature that large disordered jammed (MRJ) packings of hard frictionless spheres are isostatic [15, 16], meaning that the total number of interparticle contacts (constraints) equals the total number of degrees of freedom of system and that all of the constraints are (linearly) independent. This implies that the average number of contacts per particle Z is equal to twice the number of degrees of freedom per particle f (i.e., Z = 2 f ), in the limit as the number of particles gets large. This 3