Lowest‐density MDS array codes for reliable Smart Meter networks

A new family of maximum distance separable (MDS) array codes is presented. The code arrays contain p information columns and r independent parity columns, each column consisting of p-1 bits, where p is a prime. We extend a previously known construction for the case r=2 to three and more parity columns. It is shown that when r=3 such extension is possible for any prime p. For larger values of r, we give necessary and sufficient conditions for our codes to be MDS, and then prove that if p belongs to a certain class of primes these conditions are satisfied up to r/spl les/8. One of the advantages of the new codes is that encoding and decoding may be accomplished using simple cyclic shifts and XOR operations on the columns of the code array. We develop efficient decoding procedures for the case of two- and three-column errors. This again extends the previously known results for the case of a single-column error. Another primary advantage of our codes is related to the problem of efficient information updates. We present upper and lower bounds on the average number of parity bits which have to be updated in an MDS code over GF (2/sup m/), following an update in a single information bit. This average number is of importance in many storage applications which require frequent updates of information. We show that the upper bound obtained from our codes is close to the lower bound and, most importantly, does not depend on the size of the code symbols.

An (n, k, l) maximum distance separable (MDS) array code of length n, dimension k = n - r, and subpacketization l is formed of l × 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 ≤ 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 ≥ r <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(k-1)/r</sup> . 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 subpacketization l = r <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">⌈n-1⌉</sup> . 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 = r1.n/rl, which differs from the optimal value by at most a factor of r <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . These codes can be constructed over any finite field F as long as |F| ≥ r⌈n/r⌉, 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| ≥ n, and also afford low-complexity encoding and decoding procedures.

Maximum distance separable (MDS) codes are those codes whose minimum distance attains the Singleton bound for a given length and given dimension. It is a well-known fact that the only binary linear block codes that are MDS are the simple parity codes and the repetition codes. Known MDS codes (the Reed-Solomon codes, for instance) are defined over larger alphabets. This approach requires that (i) the encoding and decoding procedures are performed as operations over a finite field and (ii) an update in a single information bit (e.g. in storage applications) requires an update in the redundancy symbols and usually affects a number of bits in each redundancy symbol. Thus, the optimal redundancy is achieved at the expense of additional complexity in the encoding/decoding procedures as well as in the number of redundancy bits affected by an update in the information. We construct MDS codes that have the following two properties: (i) the redundancy bits are computed by simple XOR operations; and (ii) an update in an information bit affects a minimal number of redundancy bits. >

Near MDS codes of non-elliptic-curve type from Reed-Solomon codes

Maximum distance separable (MDS) codes and near MDS (NMDS) codes play a fundamental role in coding theory due to their maximal error-correcting capabilities and rich algebraic structure. This paper constructs a new class of linear codes by twice extending TGRS codes, denoted as $ C_{k,n,\lambda_{1},\lambda_{2}, \delta}(\mathit{\boldsymbol{\alpha}},\mathit{\boldsymbol{v}},\infty) $. First, we establish the necessary and sufficient conditions for $ C_{k,n,\lambda_{1},\lambda_{2}, \delta}(\mathit{\boldsymbol{\alpha}},\mathit{\boldsymbol{v}},\infty) $ to be MDS. By analyzing the AMDS properties of both $ C_{k,n,\lambda_{1},\lambda_{2}, \delta}(\mathit{\boldsymbol{\alpha}},\mathit{\boldsymbol{v}},\infty) $ and their duals $ C_{k,n,\lambda_{1},\lambda_{2}, \delta}(\mathit{\boldsymbol{\alpha}},\mathit{\boldsymbol{v}},\infty)^{\bot} $, we determine the necessary and sufficient conditions for $ C_{k,n,\lambda_{1},\lambda_{2}, \delta}(\mathit{\boldsymbol{\alpha}},\mathit{\boldsymbol{v}},\infty) $ to be NMDS. Besides, we provide the form of the parity check matrix for $ C_{k,n,\lambda_{1},\lambda_{2}, \delta}(\mathit{\boldsymbol{\alpha}},\mathit{\boldsymbol{v}},\infty) $ and characterize the weight distributions of both the codes $ C_{k,n,\lambda_{1},\lambda_{2}, \delta}(\mathit{\boldsymbol{\alpha}},\mathit{\boldsymbol{v}},\infty) $ and their duals. Additionally, using the Schur method, the non-generalized Reed-Solomon (non-GRS) properties are proven. Finally, several classes of linear complementary dual (LCD) codes are constructed.

Semi-supervised learning (SSL) has been applied to many practical applications over the past few years. Recently, distributed graph-based semi-supervised learning (DGSSL) has shown to have good performance. Traditional DGSSL algorithms usually have the problem of the straggler effect that algorithm execution time is limited by the slowest node. To solve this problem, a novel coded DGSSL(CDGSSL) algorithm based on the Maximum Distance Separable (MDS) code is proposed in this paper. Specifically, the proposed algorithm is based on the Maximum Distance Separable (MDS) code. In general, the proposed coded distributed algorithm is straggler-tolerant. Moreover, we provide optimal parameters design for the proposed algorithm. The superiority of the proposed algorithm has been confirmed via experiments on Alibaba Cloud Elastic Compute Service.

In this paper, we consider two rack-aware storage systems. First, large-scale rack-aware storage system, which is very common in large-scale storage system, is a rack-aware storage system where all sizes of racks are at least the number of redundant nodes. For such storage system, we prove that any Maximum Distance Separable (MDS) codes can have optimal inter-rack repair bandwidth and give a closed-form representation of all repair schemes with optimal inter-rack repair bandwidth. Furthermore, we show that the optimal repair access and optimal inter-rack repair bandwidth can be attained simultaneously for such storage system. Second, we investigate the rack-aware storage system of all racks with the same size, which is called uniform rack-aware storage system. We prove that, except the trivial cases, we cannot attain optimal inter-rack repair bandwidth and optimal repair access for such storage system at the same time. Specifically, we establish the lower bound of repair access for a repair scheme with optimal interrack repair bandwidth, which is tight for some parameters, and also the tight lower bound of inter-rack repair bandwidth for a repair scheme with optimal repair access.

AMDS symbol-pair codes from repeated-root cyclic codes

In this paper, we consider a novel wireless network for content distribute storage, where devices could use device-to-device (D2D) communication to transmit data. Several storage schemes, such as maximum distance separable (MDS) code and regenerating code, are employed for content downloading and repairing as node left and data lost. On the basis of multiparameter analysis, we found that targeting on the popular file caching, smaller download locality is the most suitable to the system based on fixed storage nodes and lower repair frequency. Moreover, MBR codes do not apply to D2D distributed storage due to high download cost, and MDS code is a convenient and appropriate scheme for majority practical D2D networks.

In this letter, we propose a construction of nonbinary quasi-cyclic low-density parity-check (QC-LDPC) codes based on a cyclic maximum distance separable (MDS) code. The parity-check matrices are significantly rank deficient square matrices and their Tanner graphs have a girth of at least 6. The minimum distances of the codes are very respectable as far as LDPC codes are concerned. Based on plurality voting and iterative mechanism, a low-complexity nonbinary massage-passing decoding algorithm is proposed. It only requires finite field operations, integer additions and integer comparisons. Simulation results show that the decoding algorithm is fit for the proposed codes, providing efficient trade-offs between performance and decoding complexity, which suggests that the coding scheme may find some applications in communication or storage systems with high-speed and low-power consumption requirements.

The weights in maximum distance separable (MDS) codes of length n and dimension k over the finite field GF(q) are studied. Up to some explicit exceptional cases, the MDS codes with parameters given by the MDS conjecture are shown to contain all k weights in the range n - k + 1 to n. The proof uses the covering radius of the dual code.

Determining the insdel distance of linear codes is a very challenging problem. Recently, Ji et al. [‘Strict half-Singleton bound, strict direct upper bound for linear insertion-deletion codes and optimal codes’, IEEE Trans. Inform. Theory 69 (5) (2023), 2900–2910] proposed their strict half-Singleton bound for linear codes that do not include the all-one vector. A natural question asks for linear codes with both large Hamming distance and insdel distance. Almost maximum distance separable (MDS) codes are a special kind of linear code with good Hamming error-correcting capability. We present a sufficient condition for the insdel distance of almost MDS codes to be bounded by the strict half-Singleton bound. In addition, we construct a class of two-dimensional near MDS codes that achieve the strict half-Singleton bound using twisted Reed–Solomon codes. Finally, we present a construction of optimal almost MDS insdel codes from Reed–Solomon codes for large dimensions.

The diffusion layer of the SPN block ciphers is usually built on the basis of the MDS (Maximum Distance Separable) matrices which is the matrix of the maximum distance separable code (MDS code). MDS codes have long been studied in error correcting code theory and have applications not only in coding theory but also in the design of block ciphers and hash functions. Thanks to that important role, there have been many studies on methods of building MDS matrices. In particular, the recursive MDS matrices and the symmetric recursive MDS matrices have particularly important applications because they are very efficient for execution. In this paper, we will give an estimate of the number of MDS matrices, recursive MDS matrices and symmetric recursive MDS matrices built from Reed-Solomon codes. This result is meaningful in determining the efficiency from this method of building matrices based on the Reed-Solomon codes. From there, this method can be applied to find out many MDS matrices, secure and efficient symmetric recursive MDS matrices for execution to apply in current block ciphers. Furthermore, recursive MDS matrices can be efficiently implemented using Linear Feedback Shift Registers (LFSR), making them well suited for lightweight cryptographic algorithms, so suitable for limited resources application.

The routing of vehicles or personnel in complex logistics systems is a task that needs to be solved in numerous applications, e.g., detailed models of transport networks or order picking areas. The number of relevant nodes in such networks can easily exceed 10,000 nodes. Often, a basic task is finding the shortest path from one node (start) to another (destination). Within the last years various simulation tools have been extended by respective algorithms. However, the execution time of the simulation model may significantly depend on the number of nodes in the network. We present algorithms from the literature and a comparison of three simulation tools with respect to execution time and model size for different scenarios. We further present an approach to work with so called sub-networks, i.e., the network is separated into areas, where finding the shortest path includes the task of starting at a node in one sub-network with a destination in another one.

IoT technology is widely employed to solve the problem of large-scale monitoring, e.g. in the context of smart cities or smart industries. Nevertheless, several issues have to be addressed in this context. The number of nodes can be very large and, sometimes, nodes can be not be easily reachable for humans interventions. Other important issues are battery life and node failure, two aspects that can affect the quality of service provided by the IoT system as well as related costs. To deal with the aspects above we propose a LoWPAN (Low power Wire-less Personal Area Network) network protocol that supports an automatic network construction without any human intervention. The resulting network is a tree structure featuring a main node which, in turn, is linked with the wireless gateway and a number of middle nodes (the first layer of the tree), while the leafs are called End nodes. The network structure and the underlying protocol described in this paper are designed to face the problem of configuration and to balance inter-node communication to ensure a fair power consumption. The proposed approach also features self-repair capabilities, as it is able to perform automatic recovery after a node failure. A case study is briefly discussed to show a potential application of the described approach.