a new design of binary mds array codes with asymptotically weak optimal repair binary maximum distance separable mds array codes are a special class of erasure codes for distributed storage that not only provides fault tolerance with minimum storage redundancy but also achieves low computational complexity they are constructed by encoding k information columns into r parity columns in which each element in a column is a bit such that any k out of the k r columns suffice to recover all information bits in addition to providing fault tolerance it is critical to improve repair performance in practical applications specifically if a single column fails then our goal is to minimize the repair bandwidth by downloading the least amount of bits from d healthy columns where k leq d leq k r 1 if one column of an mds code is failed it is known that we need to download at least 1 d k 1 fraction of the data stored in each of the d healthy columns if this lower bound is achieved for the repair of the failure column from accessing arbitrary d healthy columns we say that the mds code has optimal repair however if such lower bound is only achieved by d specific healthy columns then we say the mds code has weak optimal repair in this paper we propose two explicit constructions of binary mds array codes with more parity columns i e r geq 3 that achieve asymptotically weak optimal repair where k 1 leq d leq k lfloor r 1 2 rfloor and asymptotic means that the repair bandwidth achieves the minimum value asymptotically in d codes in the first construction have odd number of parity columns and asymptotically weak optimal repair for anyone information failure while codes in the second construction have even number of parity columns and asymptotically weak optimal repair for any one column failure

shortening array codes and the perfect 1 factorization conjecture the existence of a perfect 1 factorization of the complete graph with n nodes namely kn for arbitrary even number n is a 40 year old open problem in graph theory so far two infinite families of perfect 1 factorizations have been shown to exist namely the factorizations of kp 1 and k2 p where p is an arbitrary prime number p 2 it was shown in previous work that finding a perfect 1 factorization of kn is related to a problem in coding specifically it can be reduced to constructing an mds minimum distance separable lowest density array code in this paper a new method for shortening arbitrary array codes is introduced it is then used to derive the kp 1 family of perfect 1 factorization from the k2p family namely techniques from coding theory are used to prove a new result in graph theory that the two factorization families are related

new array codes for multiple phased burst correction an optimal family of array codes over gf q for correcting multiple phased burst errors and erasures where each phased burst corresponds to an erroneous or erased column in a code array is introduced as for erasures these array codes have an efficient decoding algorithm which avoids multiplications or divisions over extension fields replacing these operations with cyclic shifts of vectors over gf q the erasure decoding algorithm can be adapted easily to handle single column errors as well the codes are characterized geometrically by means of parity constraints along certain diagonal lines in each code array thus generalizing a previously known construction for the special case of two erasures algebraically they can be interpreted as reed solomon codes when q is primitive in gf q the resulting codes become conventional reed solomon codes of length p over gf q sup p 1 in which case the new erasure decoding technique can be incorporated into the berlekamp massey algorithm yielding a faster way to compute the values of any prescribed number of errors

codes with local regeneration and erasure correction regenerating codes and codes with locality are two coding schemes that have recently been proposed which in addition to ensuring data collection and reliability also enable efficient node repair in a situation where one is attempting to repair a failed node regenerating codes seek to minimize the amount of data downloaded for node repair while codes with locality attempt to minimize the number of helper nodes accessed this paper presents results in two directions in one this paper extends the notion of codes with locality so as to permit local recovery of an erased code symbol even in the presence of multiple erasures by employing local codes having minimum distance 2 an upper bound on the minimum distance of such codes is presented and codes that are optimal with respect to this bound are constructed the second direction seeks to build codes that combine the advantages of both codes with locality as well as regenerating codes these codes termed here as codes with local regeneration are codes with locality over a vector alphabet in which the local codes themselves are regenerating codes we derive an upper bound on the minimum distance of vector alphabet codes with locality for the case when their constituent local codes have a certain uniform rank accumulation property this property is possessed by both minimum storage regeneration msr and minimum bandwidth regeneration mbr codes we provide several constructions of codes with local regeneration which achieve this bound where the local codes are either msr or mbr codes also included in this paper is an upper bound on the minimum distance of a general vector code with locality as well as the performance comparison of various code constructions of fixed block length and minimum distance

x code mds array codes with optimal encoding we present a new class of mds maximum distance separable array codes of size n spl times n n a prime number called x code the x codes are of minimum column distance 3 namely they can correct either one column error or two column erasures the key novelty in x code is that it has a simple geometrical construction which achieves encoding update optimal complexity i e a change of any single information bit affects exactly two parity bits the key idea in our constructions is that all parity symbols are placed in rows rather than columns

quasinormal cones in banach spaces keywords quasinormality cone in banach space reference ana article 1983 001doi 10 1016 0362 546x 83 90028 7view record in web of science record created on 2008 12 10 modified on 2016 08 08

repairing reed solomon codes we study the performance of reed solomon rs codes for the exact repair problem in distributed storage our main result is that in some parameter regimes reed solomon codes are optimal regenerating codes among maximum distance separable mds codes with linear repair schemes moreover we give a characterization of mds codes with linear repair schemes which holds in any parameter regime and which can be used to give non trivial repair schemes for rs codes in other settings more precisely we show that for k dimensional rs codes whose evaluation points are a finite field of size n there are exact repair schemes with bandwidth n 1 log n 1 n k bits and that this is optimal for any mds code with a linear repair scheme in contrast the naive commonly implemented repair algorithm for this rs code has bandwidth k log n bits when the entire field is used as evaluation points the number of nodes n is much larger than the number of bits per node which is o log n and so this result holds only when the degree of sub packetization is small however our method applies in any parameter regime and to illustrate this for high levels of sub packetization we give an improved repair scheme for a specific 14 10 rs code used in the facebook hadoop analytics cluster

regular article a fast adaptive multipole algorithm in three dimensions we present an adaptive fast multipole method for the laplace equation in three dimensions it uses both new compression techniques and diagonal forms for translation operators to achieve high accuracy at a reasonable cost

explicit constructions of optimal access mds codes with nearly optimal sub packetization an n k l maximum distance separable mds array code of length n dimension k n r and sub packetization l is formed of l times n matrices over a finite field f with every column of the matrix stored on a separate node in the distributed storage system and viewed as a coordinate of the codeword repair of a failed node recovery of one erased column can be performed by accessing a set of d le n 1 surviving helper nodes the code is said to have the optimal access property if the amount of data accessed at each of the helper nodes meets a lower bound on this quantity for optimal access mds codes with d n 1 the sub packetization l satisfies the bound l ge r k 1 r in our previous work ieee trans inf theory vol 63 no 4 2017 for any n and r we presented an explicit construction of optimal access mds codes with sub packetization l r n 1 in this paper we take up the question of reducing the sub packetization value l to make it to approach the lower bound we construct an explicit family of optimal access codes with l r lceil n r rceil which differs from the optimal value by at most a factor of r 2 these codes can be constructed over any finite field f as long as f ge r lceil n r rceil and afford low complexity encoding and decoding procedures we also define a version of the repair problem that bridges the context of regenerating codes and codes with locality constraints lrc codes which we call group repair with optimal access in this variation we assume that the set of n sm nodes is partitioned into m repair groups of size s and require that the amount of accessed data for repair is the smallest possible whenever the d s k 1 helper nodes include all the other s 1 nodes from the same group as the failed node for this problem we construct a family of codes with the group optimal access property these codes can be constructed over any field f of size f ge n and also afford low complexity encoding and decoding procedures

an encoding algorithm of triply extended reed solomon codes with asymptotically optimal complexities in this paper we devise a fast encoding algorithm for triply extended reed solomon codes the proposed approach requires approximately two xors per bit which improves the prior result of three xors per bit established by certain maximum distance separable mds array codes we also prove that for mds codes with two and three parities the scheduling algorithms require at least two xors per bit to the best of our knowledge this is the first provable scheduling algorithm for the triple parity mds codes to approach the theoretical lower bounds the implementation with simd instructions is provided the simulations show that the proposed approach is competitive as compared with other cutting edge implementations

constructions of high rate minimum storage regenerating codes over small fields a novel technique for construction of minimum storage regenerating msr codes is presented based on this technique three explicit constructions of msr codes are given the first two constructions provide access optimal msr codes with two and three parities respectively which attain the sub packetization bound for access optimal codes the third construction provides longer msr codes with three parities i e codes with larger number of systematic nodes this improvement is achieved at the expense of the access optimality and the field size in addition to a minimum storage in a node all three constructions allow the entire data to be recovered from a minimal number of storage nodes that is given storage ell in each node the entire stored data can be recovered from any 2 log 2 ell for two parity nodes and either 3 log 3 ell or 4 log 3 ell for three parities second in the first two constructions a helper node accesses the minimum number of its symbols for repair of a failed node access optimality the goal of this paper is to provide a construction of such optimal codes over the smallest possible finite fields the generator matrix of these codes is based on perfect matchings of complete graphs and hypergraphs and on a rational canonical form of matrices for two parities the field size is reduced by a factor of two for access optimal codes compared to previous constructions for three parities in the first construction a field size of at least 6 log 3 ell 1 or 3 log 3 ell 1 for fields with characteristic 2 is sufficient and in the second construction the field size is larger yet linear in log 3 ell both constructions with three parities provide a significant improvement over previous works due to either decreased field size or lower subpacketization

higher order adaptive finite element methods for kohn sham density functional theory we present an efficient computational approach to perform real space electronic structure calculations using an adaptive higher order finite element discretization of kohn sham density functional theory dft to this end we develop an a priori mesh adaption technique to construct a close to optimal finite element discretization of the problem we further propose an efficient solution strategy for solving the discrete eigenvalue problem by using spectral finite elements in conjunction with gauss lobatto quadrature and a chebyshev acceleration technique for computing the occupied eigenspace the proposed approach has been observed to provide a staggering 100 200 fold computational advantage over the solution of a generalized eigenvalue problem using the proposed solution procedure we investigate the computational efficiency afforded by higher order finite element discretizations of the kohn sham dft problem our studies suggest that staggering computational savings of the order of 1000 fold relative to linear finite elements can be realized for both all electron and local pseudopotential calculations by using higher order finite element discretizations on all the benchmark systems studied we observe diminishing returns in computational savings beyond the sixth order for accuracies commensurate with chemical accuracy suggesting that the hexic spectral element may be an optimal choice for the finite element discretization of the kohn sham dft problem a comparative study of the computational efficiency of the proposed higher order finite element discretizations suggests that the performance of finite element basis is competing with the plane wave discretization for non periodic local pseudopotential calculations and compares to the gaussian basis for all electron calculations to within an order of magnitude further we demonstrate the capability of the proposed approach to compute the electronic structure of a metallic system containing 1688 atoms using modest computational resources and good scalability of the present implementation up to 192 processors

atom partitioned multipole expansions for electrostatic potential boundary conditions applications such as grid based real space density functional theory dft use the poisson equation to compute electrostatics however the expected long tail of the electrostatic potential requires either the use of a large and costly outer domain or dirichlet boundary conditions estimated via multipole expansion we find that the oft used single center spherical multipole expansion is only appropriate for isotropic mesh domains such as spheres and cubes in this work we introduce a method suitable for high aspect ratio meshes whereby the charge density is partitioned into atomic domains and multipoles are computed for each domain while this approach is moderately more expensive than a single center expansion it is numerically stable and still a small fraction of the overall cost of a dft calculation the net result is that when high aspect ratio systems are being studied form fitted meshes can now be used in lieu of cubic meshes to gain computational speedup

on lowest density mds codes let f sub q denote the finite field gf q and let h be a positive integer mds maximum distance separable codes over the symbol alphabet f sub q sup b are considered that are linear over f sub q and have sparse low density parity check and generator matrices over f sub q that are systematic over f sub q sup b lower bounds are presented on the number of nonzero elements in any systematic parity check or generator matrix of an f sub q linear mds code over f sub q sup b along with upper bounds on the length of any mds code that attains those lower bounds a construction is presented that achieves those bounds for certain redundancy values the building block of the construction is a set of sparse nonsingular matrices over f sub q whose pairwise differences are also nonsingular bounds and constructions are presented also for the case where the systematic condition on the parity check and generator matrices is relaxed to be over f sub q rather than over f sub q sup b

maximizing burst erasure correction capability of mds codes in general an n k maximum distance separable mds code over gf pm can not correct all burst erasures of length n k m over gf p in this letter we constructively show that such a linear mds code over gf pm can be modified to correct all burst erasures of length up to n k m over gf p while maintaining its mds property over gf pm

new efficient mds array codes for raid part ii rabin like codes for tolerating multiple spl ges 4 disk failures for pt 1 see ibid vol 54 no 9 p 1071 1080 2005 a new class of binary maximum distance separable mds array codes which are based on circular permutation matrices are introduced in this paper these array codes are used for tolerating multiple spl ges 4 disk failures in redundant arrays of inexpensive disks raid architecture the size of the information part is m spl times n where n is the number of information disks and m 1 is a prime integer the size of the parity check part is m spl times r the minimum distance is r 1 and the number of parity check disks is r in practical applications m can be very large and n ranges from 20 to 50 the code rate is r n n r these codes can be used for tolerating up to r disk failures with very fast encoding and decoding the complexities of encoding and decoding algorithms are o rmn and o m sup 3 r sup 4 respectively when r 4 there need to be 9mn xor operations for encoding and 9n 95 m 1 xor operations for decoding

new efficient mds array codes for raid part i reed solomon like codes for tolerating three disk failures this paper presents a class of binary maximum distance separable mds array codes for tolerating disk failures in redundant arrays of inexpensive disks raid architecture based on circular permutation matrices the size of the information part is m spl times n the size of the parity check part is m spl times 3 and the minimum distance is 4 where n is the number of information disks the number of parity check disks is 3 and m 1 is a prime integer in practical applications m can be very large and n is from 20 to 50 the code rate is r n n 3 these codes can be used for tolerating three disk failures the encoding and decoding of the reed solomon like codes are very fast there need to be 3mn xor operations for encoding and 3mn 9 m 1 xor operations for decoding

two classes of binary mds array codes with asymptotically optimal repair for any single column an m times k r binary maximum distance separable mds array code contains k information columns and r parity columns with each entry being a bit where any k out of k r columns can recover the k information columns when there is a failed column it is critical to minimize the repair bandwidth that is the total number of bits downloaded from d out of k r 1 surviving columns in repairing the failed column in this article we first propose two explicit constructions of binary mds array codes that have asymptotically optimal repair bandwidth for any information column where r geq 2 and d k r 1 for the first construction and r geq 4 is an even number and d k frac r 2 1 for the second construction by applying a generic transformation for the proposed two classes of binary mds array codes we then obtain two classes of new binary mds array codes that also have optimal repair bandwidth for any parity column and asymptotically optimal repair bandwidth for any information column

xi code a family of practical lowest density mds array codes of distance 4 designing the lowest density maximum distance separable mds array codes has gained much attention in recent years due to the optimal redundancy and minimum update penalty lowest density of such codes however the existing lowest density mds array codes of distance 4 have extremely strict constraints on the code length which makes them impractical in particular most of them require the code length to be p or p 1 where p is a prime that satisfies i 3 p 1 and ii 2 is primitive in mathrm gf p in this paper we propose a new family of the lowest density mds array codes of distance 4 called xi code it has the properties of 1 being capable of correcting both triple erasures and a single error combined with one erasure 2 having code length of either p or p 1 with p being an odd prime number 3 achieving optimality in encoding and update and 4 achieving optimality in erasure decoding for certain erasure patterns and near optimality for other erasure patterns it is worth mentioning that xi code is the first discovered family of the lowest density mds array codes of distance 4 that supports the code length of p or p 1 with p being an odd prime

cyclic lowest density mds array codes three new families of lowest density maximum distance separable mds array codes are constructed which are cyclic or quasi cyclic in addition to their optimal redundancy mds and optimal update complexity lowest density the symmetry offered by the new codes can be utilized for simplified implementation in storage applications the proof of the code properties has an indirect structure first mds codes that are not cyclic are constructed and then transformed to cyclic codes by a minimum distance preserving transformation

phased burst error correcting array codes various aspects of single phased burst error correcting array codes are explored these codes are composed of two dimensional arrays with row and column parities with a diagonally cyclic readout order they are capable of correcting a single burst error along one diagonal optimal codeword sizes are found to have dimensions n sub 1 n sub 2 such that n sub 2 is the smallest prime number larger than n sub 1 these codes are capable of reaching the singleton bound a new type of error approximate errors is defined in q ary applications these errors cause data to be slightly corrupted and therefore still close to the true data level phased burst array codes can be tailored to correct these codes with even higher rates than before

higher order adaptive finite element methods for orbital free density functional theory in the present work we study various numerical aspects of higher order finite element discretizations of the non linear saddle point formulation of orbital free density functional theory we first investigate the robustness of viable solution schemes by analyzing the solvability conditions of the discrete problem we find that a staggered solution procedure where the potential fields are computed consistently for every trial electron density is a robust solution procedure for higher order finite element discretizations we next study the convergence properties of higher order finite element discretizations of orbital free density functional theory by considering benchmark problems that include calculations involving both pseudopotential as well as coulomb singular potential fields our numerical studies suggest close to optimal rates of convergence on all benchmark problems for various orders of finite element approximations considered in the present study we finally investigate the computational efficiency afforded by various higher order finite element discretizations which constitutes the main aspect of the present work by measuring the cpu time for the solution of discrete equations on benchmark problems that include large aluminum clusters in these studies we use mesh coarse graining rates that are derived from error estimates and an a priori knowledge of the asymptotic solution of the far field electronic fields our studies reveal a significant 100 1000 fold computational savings afforded by the use of higher order finite element discretization alongside providing the desired chemical accuracy we consider this study as a step towards developing a robust and computationally efficient discretization of electronic structure calculations using the finite element basis

access versus bandwidth in codes for storage maximum distance separable mds codes are widely used in storage systems to protect against disk node failures a node is said to have capacity l over some field f if it can store that amount of symbols of the field an n k l mds code uses n nodes of capacity l to store k information nodes the mds property guarantees the resiliency to any n k node failures an optimal bandwidth respectively optimal access mds code communicates respectively accesses the minimum amount of data during the repair process of a single failed node it was shown that this amount equals a fraction of 1 n k of data stored in each node in previous optimal bandwidth constructions l scaled polynomially with k in codes when the asymptotic rate is less than 1 moreover in constructions with a constant number of parities i e when the rate approaches 1 l is scaled exponentially with k in this paper we focus on the case of linear codes with linear repair operations and constant number of parities n k r and ask the following question given the capacity of a node l what is the largest number of information disks k in an optimal bandwidth respectively access k r k l mds code we give an upper bound for the general case and two tight bounds in the special cases of two important families of codes the first is a family of codes with optimal update property and the second is a family with optimal access property moreover the bounds show that in some cases optimal bandwidth codes have larger k than optimal access codes and therefore these two measures are not equivalent