Codes With Hierarchical Locality

In this paper, we study the notion of {\em codes with hierarchical locality} that is identified as another approach to local recovery from multiple erasures. The well-known class of {\em codes with locality} is said to possess hierarchical locality with a single level. In a {\em code with two-level hierarchical locality}, every symbol is protected by an inner-most local code, and another middle-level code of larger dimension containing the local code. We first consider codes with two levels of hierarchical locality, derive an upper bound on the minimum distance, and provide optimal code constructions of low field-size under certain parameter sets. Subsequently, we generalize both the bound and the constructions to hierarchical locality of arbitrary levels.Comment: 12 pages, submitted to ISIT 201

A family of optimal locally recoverable codes

A code over a finite alphabet is called locally recoverable (LRC) if every symbol in the encoding is a function of a small number (at most $r$) other symbols. We present a family of LRC codes that attain the maximum possible value of the distance for a given locality parameter and code cardinality. The codewords are obtained as evaluations of specially constructed polynomials over a finite field, and reduce to a Reed-Solomon code if the locality parameter $r$ is set to be equal to the code dimension. The size of the code alphabet for most parameters is only slightly greater than the code length. The recovery procedure is performed by polynomial interpolation over $r$ points. We also construct codes with several disjoint recovering sets for every symbol. This construction enables the system to conduct several independent and simultaneous recovery processes of a specific symbol by accessing different parts of the codeword. This property enables high availability of frequently accessed data ("hot data").Comment: Minor changes. This is the final published version of the pape

Codes with efficient erasure correction

Distributed storage systems are becoming increasingly ubiquitous in the emerging era of Internet of Things. Major internet technology companies employ large-scale distributed storage systems to accommodate the massive amounts of data generated and requested by global users. The need of reliable and efficient storage of immense amounts of data calls for new applications and development of classical error-correcting codes. This dissertation is devoted to a study of codes with efficient erasure correction for distributed storage systems. The efficiency of erasure correction is often assessed by two performance metrics, bandwidth and locality. In this dissertation we address several problems for each of these two metrics. We construct families of codes with optimal communication complexity for erasure correction ("repair bandwidth") for a heterogeneous storage model, and derive several results for the problem of optimal repair of Reed-Solomon codes. We also construct families of cyclic and convolutional codes with locality, extending the range of parameters for which such families were previously known

On subspace designs

Guruswami and Xing introduced subspace designs in 2013 to give the first construction of positive rate rank metric codes list-decodable beyond half the distance. In this paper we provide bounds involving the parameters of a subspace design, showing they are tight via explicit constructions. We point out a connection with sum-rank metric codes, dealing with optimal codes and minimal codes with respect to this metric. Applications to two-intersection sets with respect to hyperplanes, two-weight codes, cutting blocking sets and lossless dimension expanders are also provided

Error-Correcting Codes for Networks, Storage and Computation

The advent of the information age has bestowed upon us three challenges related to the way we deal with data. Firstly, there is an unprecedented demand for transmitting data at high rates. Secondly, the massive amounts of data being collected from various sources needs to be stored across time. Thirdly, there is a need to process the data collected and perform computations on it in order to extract meaningful information out of it. The interconnected nature of modern systems designed to perform these tasks has unraveled new difficulties when it comes to ensuring their resilience against sources of performance degradation. In the context of network communication and distributed data storage, system-level noise and adversarial errors have to be combated with efficient error correction schemes. In the case of distributed computation, the heterogeneous nature of computing clusters can potentially diminish the speedups promised by parallel algorithms, calling for schemes that mitigate the effect of slow machines and communication delay. This thesis addresses the problem of designing efficient fault tolerance schemes for the three scenarios just described. In the network communication setting, a family of multiple-source multicast networks that employ linear network coding is considered for which capacity-achieving distributed error-correcting codes, based on classical algebraic constructions, are designed. The codes require no coordination between the source nodes and are end to end: except for the source nodes and the destination node, the operation of the network remains unchanged. In the context of data storage, balanced error-correcting codes are constructed so that the encoding effort required is balanced out across the storage nodes. In particular, it is shown that for a fixed row weight, any cyclic Reed-Solomon code possesses a generator matrix in which the number of nonzeros is the same across the columns. In the balanced and sparsest case, where each row of the generator matrix is a minimum distance codeword, the maximal encoding time over the storage nodes is minimized, a property that is appealing in write-intensive settings. Analogous constructions are presented for a locally recoverable code construction due to Tamo and Barg. Lastly, the problem of mitigating stragglers in a distributed computation setup is addressed, where a function of some dataset is computed in parallel. Using Reed-Solomon coding techniques, a scheme is proposed that allows for the recovery of the function under consideration from the minimum number of machines possible. The only assumption made on the function is that it is additively separable, which renders the scheme useful in distributed gradient descent implementations. Furthermore, a theoretical model for the run time of the scheme is presented. When the return time of the machines is modeled probabilistically, the model can be used to optimally pick the scheme's parameters so that the expected computation time is minimized. The recovery is performed using an algorithm that runs in quadratic time and linear space, a notable improvement compared to state-of-the-art schemes. The unifying theme of the three scenarios is the construction of error-correcting codes whose encoding functions adhere to certain constraints. It is shown that in many cases, these constraints can be satisfied by classical constructions. As a result, the schemes presented are deterministic, operate over small finite fields and can be decoded using efficient algorithms.</p

Design and Performance Evaluation of Nonlinear Collimation Systems for CLIC and LHC

The beam collimation systems are an essential part of the high energy colliders. A collimation system should remove beam halo to reduce detector background and ensure the machine protection, thus minimizing the activation and damage of sensitive accelerator components. The mechanical and optics design of collimation systems is not simple, and they should fullfil some often conflicting constraints and requirements: high cleaning efficiency, high mechanical robustness, and low wakefields (impedances). The conventional collimation systems are generally based on linear optics. Nevertheless, several alternative advanced concepts on collimation have been proposed in the literature. In this thesis report we have studied in detail nonlinear collimation systems. These are based on a general scheme with a skew sextupole pair, which can be adapted to both linear and circular colliders. In particular we have designed a nonlinear collimation system for the Compact Linear Collider (CLIC). This system fullfils the function of machine protection against mis-steered or errant beams with energy offset higher than 1.5 % of the nonimal energy 1.5 TeV. The performance of this collimation system has been evaluated by means of tracking studies, and compared with that of the conventional baseline linear collimation system. Since the collimation requirements for linear colliders designed to operate at center-of-mass energy around TeV are similar to tho se for the Large Hadron Collider (LHC) at collision beam energy 7 TeV, it is thus close thought to apply a similar LHC nonlinear collimation scheme as that designed for CLIC. We have explored this possibility, and have proposed an alternative nonlinear system for the Phase-II betatron cleaning in the LHC. Its performance and cleaning efficiency have further been evaluated by tracking studies. Moreover a comparison of the features of the nonlinear collimation system and the linear collimation system has been made for the LHC

BOLTED CONNECTIONS FOR EASILY REPAIRABLE SEISMIC RESISTANT STEEL STRUCTURES

Recent years have brought significant advances in the design capabilities and construction practices of steel structures. These were partially caused by technological development and a direct effect of the research community efforts towards the mitigation of the earthquake induced damage. Making the traditional structural systems more resilient is one of the directions taken but, more and more, solutions with reduced post-earthquake repair costs are preferred. Steel structures are particularly malleable in the modern spirit of integrating devices which render the structure as “low-damage” or “easily repairable”. The recent earthquakes of Japan and New Zealand have demonstrated the feasibility and the advantages of such structural typologies. The current work presents an investigation on two steel structural solutions, including thus both moment resisting and braced frames, which have the potential of being easily used in practice, with minimal alteration of the design and erection procedures and improved post-earthquake economic benefits. The thesis focuses on (i) bolted connections of detachable short links for eccentrically braced frame and second, and (ii) on bolted friction connections for moment resisting frames. The main objective is to facilitate the application of these structural solutions in practice by enhancing the knowledge of their relevant bolted connection design and behavior