Explicit and Optimal Exact-Regenerating Codes for the Minimum-Bandwidth Point in Distributed Storage

In the distributed storage setting that we consider, data is stored across n nodes in the network such that the data can be recovered by connecting to any subset of k nodes. Additionally, one can repair a failed node by connecting to any d nodes while downloading beta units of data from each. Dimakis et al. show that the repair bandwidth d beta can be considerably reduced if each node stores slightly more than the minimum required and characterize the tradeoff between the amount of storage per node and the repair bandwidth. In the exact regeneration variation, unlike the functional regeneration, the replacement for a failed node is required to store data identical to that in the failed node. This greatly reduces the complexity of system maintenance. The main result of this paper is an explicit construction of codes for all values of the system parameters at one of the two most important and extreme points of the tradeoff - the Minimum Bandwidth Regenerating point, which performs optimal exact regeneration of any failed node. A second result is a non-existence proof showing that with one possible exception, no other point on the tradeoff can be achieved for exact regeneration.

A Flexible Class of Regenerating Codes for Distributed Storage

In the distributed storage setting introduced by Dimakis et al., B units of data are stored across n nodes in the network in such a way that the data can be recovered by connecting to any k nodes. Additionally one can repair a failed node by connecting to any d nodes while downloading at most beta units of data from each node. In this paper, we introduce a flexible framework in which the data can be recovered by connecting to any number of nodes as long as the total amount of data downloaded is at least B. Similarly, regeneration of a failed node is possible if the new node connects to the network using links whose individual capacity is bounded above by beta(max) and whose sum capacity equals or exceeds a predetermined parameter gamma. In this flexible setting, we obtain the cut-set lower bound on the repair bandwidth along with a constructive proof for the existence of codes meeting this bound for all values of the parameters. An explicit code construction is provided which is optimal in certain parameter regimes.

On Network-Error Correcting Convolutional Codes under the BSC Edge Error Model

Convolutional network-error correcting codes (CNECCs) are known to provide error correcting capability in acyclic instantaneous networks within the network coding paradigm under small field size conditions. In this work, we investigate the performance of CNECCs under the error model of the network where the edges are assumed to be statistically independent binary symmetric channels, each with the same probability of error pe(0 <= p(e) < 0.5). We obtain bounds on the performance of such CNECCs based on a modified generating function (the transfer function) of the CNECCs. For a given network, we derive a mathematical condition on how small p(e) should be so that only single edge network-errors need to be accounted for, thus reducing the complexity of evaluating the probability of error of any CNECC. Simulations indicate that convolutional codes are required to possess different properties to achieve good performance in low p(e) and high p(e) regimes. For the low p(e) regime, convolutional codes with good distance properties show good performance. For the high p(e) regime, convolutional codes that have a good slope ( the minimum normalized cycle weight) are seen to be good. We derive a lower bound on the slope of any rate b/c convolutional code with a certain degree.

Queued Cooperative Wireless Networks With Rateless Codes

Cooperative communication using rateless codes, in which the source transmits an infinite number of parity bits to the destination until the receipt of an acknowledgment, has recently attracted considerable interest. It provides a natural and efficient mechanism for accumulating mutual information from multiple transmitting relays. We develop an analysis of queued cooperative relay systems that combines the communication-theoretic transmission aspects of cooperative communication using rateless codes over Rayleigh fading channels with the queuing-theoretic aspects associated with buffering messages at the relays. Relay cooperation combined with queuing reduces the message transmission times and also helps distribute the traffic load in the network, which improves throughput significantly.

Code Generation Paradigms On Optical Orthogonal And Other Codes In a Fiber Optic CDMA Lan

Code Division Multiple Access (CDMA) techniques, by far, had been applied to LAN problems by many investigators, An analytical study of well known algorithms for generation of Orthogonal codes used in FO-CDMA systems like those for prime, quasi-Prime, Optical Orthogonal and Matrix codes has been presented, Algorithms for OOCs like Greedy/Modified Greedy/Accelerated Greedy algorithms are implemented. Many speed-up enhancements. for these algorithms are suggested. A novel Synthetic Algorithm based on Difference Sets (SADS) is also proposed. Investigations are made to vectorise/parallelise SADS to implement the source code on parallel machines. A new matrix for code families of OOCs with different seed code-words but having the same (n,w,lambda) set is formulated.

Derivation and consistency of the partial functions of the ternary system involving interaction coefficients

The logarithm of activity coefficients of the components of the ternary system is derived based on the Maclaurin infinite series, which is expressed in terms of the integral property of the system and subjected to appropriate boundary conditions. The derivation of the functions involves extensive summation of various infinite series pertaining to the first-order interaction coefficients that have been shown completely to remove any truncational error. Since the conventional equations involving interaction coefficients are internally inconsistent, a consistent form of the partial functions is developed in the article using the technique just described. The thermodynamic consistency of the functions based on the Maxwell and the Gibbs-Duhem relations has been established. The derived values of the logarithmic activity coefficients of the components have been found to be in agreement with the thermodynamic data of the Fe-Cr-Ni system at 1873 K and have been found to be independent of the compositional paths.

Representation of thermodynamic properties of ternary systems and its application to the system silver-gold-copper at 1350 K

The article presents a generalized analytical expression for description of the integral excess Gibbs free energy of mixing of a ternary system. Twelve constants of the equation are assessed by the least mean squares regressional analysis of the experimental integral excess data of the constituent binaries; three ternary parameters are evaluated by a regressional analysis based on the partial experimental data of a component of the ternary system. The assessed values of the ternary parameters describe the nature of the ternary interaction in the system. Activities and isoactivities of the components in the Ag-Au-Cu system at 1350 K are calculated and found to be in good agreement with the experimental data. This analytical treatment is particularly useful to ternary systems where the thermodynamic data are available from different sources.

Solid state amorphization in binary Ti---Ni, Ti---Cu and ternary Ti---Ni---Cu system by mechanical alloying

An amorphous phase has been synthesized by mechanical alloying in a planetary mill over a nickel content range of 10�70 at.% in the Ti---Ni system and a copper content range of 10�50 at.% in the Ti---Cu system. In the case of ternary Ti---Ni---Cu alloys the glass-forming composition range has been found to be given by x = 10�20 for Ti60Ni40 ? xCux, x = 10 � 30 for Ti50Ni50 ? xCux and x = 10 � 40 for Ti40Ni60 ? xCux alloys. The difficulty in the amorphization of copper-rich compositions is explained in the light of enthalpy composition diagrams calculated for the ternary solid solution and the amorphous phase.

Study of the reactions of pyridinium poly(hydrogen fluoride) with some ternary oxides

The products of the reaction of pyridinium poly(hydrogen fluoride), PPHF, with KIO3, Na2SnO3, NaBiO3, K2CrO4, Na2MoO4 and Na2WO4 were KIO2F2; Na2SnF6; NaHF2, BiF3; K3CrF6, KHF2, (PyH)(3)CrF6; NaHF2, (PyH)(2)MoO2F4.2NaHF(2); and (PyH)(2)WO2F4.2NaHF(2), respectively, while KClO3, KBrO3 and KlO(4) react with complete decomposition to form KHF2 as the fluorinated product. This differential reactivity and mode of reaction has been discussed in terms of the oxidation state of the central atom, the nature and strength of the bonds and the complex behaviour of the formed intermediate or fluorinated products that undergo complexation or solvation with pyridine and/or hydrogen fluoride.

Unusual ribose conformation in a ternary copper(II) 2?deoxyribonucleotide complex

The ternary metal deoxyribonucleotide complex [Cu(bzim)(5?-dGMP)(H2O)3](bzim = benzimidazole, 5?-dGMP = 2?-deoxyguanosine 5?-monophosphate) has been prepared and the structure analysed by X-ray diffraction. The compound crystallizes in the space group P1 with a= 7.069(6), b= 13.959(10), c= 14.204(12)Å, ?= 75.12(6), ?= 94.15(6), ?= 97.98(6)° and Z= 2. The structure was solved by the heavy-atom method and refined by full-matrix least-squares procedures on the basis of 2813 observed [I[gt-or-equal] 3?(I)] reflections to final R and R? values of 0.050 and 0.052 respectively. There are two independent molecules in the asymmetric unit and both copper(II) centres have square-pyramidal co-ordination geometry. An unusual feature of the structure is the co-ordination of the metal by N(7) of the base, in the presence of a ?-aromatic amine, bzim. The structure is stabilized by intermolecular base�bzim stacking. The nucleotides of both the molecules have an anti conformation about the glycosyl bond, and a gauche-gauche conformation about the C(4?)�C(5?) bond. A feature of particular interest is the unusual sugar conformation. The base furanose rings of the two nucleotide molecules adopt C(3?)-exo/C(2?)-endo pucker and C(3?)-exo pucker respectively.

MIMO Precoding With X- and Y- Codes

We consider a slow fading multiple-input multiple-output (MIMO) system with channel state information at both the transmitter and receiver. A well-known precoding scheme is based upon the singular value decomposition (SVD) of the channel matrix, which transforms the MIMO channel into parallel subchannels. Despite having low maximum likelihood decoding (MLD) complexity, this SVD precoding scheme provides a diversity gain which is limited by the diversity gain of the weakest subchannel. We therefore propose X- and Y-Codes, which improve the diversity gain of the SVD precoding scheme but maintain the low MLD complexity, by jointly coding information across a pair of subchannels. In particular, subchannels with high diversity gain are paired with those having low diversity gain. A pair of subchannels is jointly encoded using a 2 2 real matrix, which is fixed a priori and does not change with each channel realization. For X-Codes, these rotation matrices are parameterized by a single angle, while for Y-Codes, these matrices are left triangular matrices. Moreover, we propose X-, Y-Precoders with the same structure as X-, Y-Codes, but with encoding matrices adapted to each channel realization. We observed that X-Codes/Precoders are good for well-conditioned channels, while Y-Codes/Precoders are good for ill-conditioned channels.

Relation between integral and partial properties of ternary solutions along different composition trajectories

The relations between partial and integral properties of ternary solutions along composition trajectories suggested by Kohler, Colinet and Jacob, and along an arbitrary path are derived. The chemical potentials of the components are related to the slope of integral free energy by expressions involving the binary compositions generated by the intersections of the composition trajectory with the sides of the ternary triangle. Only along the Kohler composition trajectory it is possible to derive the integral free energy from the variation of the chemical potential of a single component with composition or vice versa. Along all other paths the differential of the integral free energy is related to two chemical potentials. The Gibbs-Duhem integration proposed by Darken for the ternary system uses the Kohler isogram. The relative merits of different limits for integration are discussed.

Stabilities of ternary carbides UWC1.75 and UWC2

The methane-hydrogen gas equilibration technique has been used to measure the chemical potential of carbon associated with two three-phase fields of the system U-W-C in the temperature range 973 to 1173 K. By combining the values of the chemical potential of carbon in the three-phase fields UC + W + UWC1.75 and UC + UWC1.75 + UWC2 Obtained in this study with the data on the Gibbs energy of formation of UC available in the literature, expressions for the Gibbs energies of formation of the two ternary carbides were derived: Delta(f)G degrees [UWC1.75] = -131, 600 - 300 T (+/-8000) J mol(-1) Delta(f)G degrees [UWC2] = -144, 800 - 32.0 T (+/- 10,000) J mol(-1) Although estimates of Gibbs energies of formation of the two ternary carbides TSWC1.75 and UWC2 have been reported, there have been no previous experimental determinations of thermodynamic properties of these compounds.

Thermodynamic constraints along isograms in ternary systems: application to three-phase equilibria

Thermodynamic constraints on component chemical potentials in three-phase fields introduced by the various isograms suggested in the literature are derived for a ternary system containing compounds. When compositions of two compounds lie on an isogram, it is associated with specific characteristics which can be used to obtain further understanding of the interplay of thermodynamic factors that determine phase equilibria. When two compounds are shared by adjacent three-phase fields, the constraints are dictated by binary compositions generated by the intersection of a line passing through the shared compounds with the sides of the ternary triangle. Generalized expressions for an arbitrary line through the triangle are presented. These are consistent with special relations obtained along Kohler, Colinet and Jacob isograms. Five axioms are introduced and proved. They provide valuable tools for checking consistency of thermodynamic measurements and for deriving thermodynamic properties from phase diagrams. (C) 1997 Elsevier Science S.A.