Two-layer Locally Repairable Codes for Distributed Storage Systems

In this paper, we propose locally repairable codes (LRCs) with optimal minimum distance for distributed storage systems (DSS). A two-layer encoding structure is employed to ensure data reconstruction and the designated repair locality. The data is first encoded in the first layer by any existing maximum distance separable (MDS) codes, and then the encoded symbols are divided into non-overlapping groups and encoded by an MDS array code in the second layer. The encoding in the second layer provides enough redundancy for local repair, while the overall code performs recovery of the data based on redundancy from both layers. Our codes can be constructed over a finite field with size growing linearly with the total number of nodes in the DSS, and facilitate efficient degraded reads.Comment: This paper has been withdrawn by the author due to inaccuracy of Claim

Capabilities' Substitutability and the "S" Curve of Export Diversity

Product diversity, which is highly important in economic systems, has been highlighted by recent studies on international trade. We found an empirical pattern, designated as the "S-shaped curve", that models the relationship between economic size (logarithmic GDP) and export diversity (the number of varieties of export products) on the detailed international trade data. As the economic size of a country begins to increase, its export diversity initially increases in an exponential manner, but overtime, this diversity growth slows and eventually reaches an upper limit. The interdependence between size and diversity takes the shape of an S-shaped curve that an be fitted by a logistic equation. To explain this phenomenon, we introduce a new parameter called "substitutability" into the list of capabilities or factors of products in the tri-partite network model (i.e., the country-capability-product model) of Hidalgo et al. As we observe, when the substitutability is zero, the model returns to Hidalgo's original model but failed to reproduce the S-shaped curve. However, in a plot of data, the data increasingly resembles an the S-shaped curve as the substitutability expands. Therefore, the diversity ceiling effect can be explained by the substitutability of different capabilities

Combined X-ray diffraction and diffuse reflectance analysis of nanocrystalline mixed Sn(II) and Sn(IV) Oxide Powders

Nanocrystalline mixtures of Sn(II) and Sn(IV) oxide powders, potential gas sensor materials, are synthesized via a simple precipitation route using SnCl(2) as the precursor. Materials are characterized by powder X-ray diffraction, thermogravimetric analysis, UV-visible diffuse reflectance spectroscopy (DRS), and Fourier transform infrared spectroscopy. The ratio of Sn(II)/Sn(IV) in powders precipitated at room temperature, as well as the identity of the primary Sn(II) product (SnO or Sn(6)O(4)(OH)(4)), can be varied by adjusting aging time and washing procedures. The identity of the initial Sn(II) product influences the subsequent phase composition and degree of disorder in the tetragonal SnO(2) phase obtained following sintering in air. Analysis of the DRS absorption edge and long-wavelength (Urbach) absorption tail is used to determine the SnO(2) optical band gap and extent of disorder. SnO(2) obtained by heating the SnO/SnO(2) mixture at 600 or 800 degrees C has a smaller optical band gap and a broader Urbach tail than the analogous sample obtained from heating Sn(6)O(4)(OH)(4), indicating a more disordered material