Akademik Veri Yönetim Sistemi
KÖROĞLU M. E. (Yürütücü), ZENGİN R.
Yükseköğretim Kurumları Destekli Proje, 2023 - 2024
Distributed and cloud storage systems are widely utilized to store large amounts of data in modern data centers such as Windows Azure Storage and Hadoop Distributed File System RAID utilized by Facebook. Storage node failures may be encountered in these systems. Thus, these systems have to provide high data reliability and availability. For this purpose, redundant data are used in large distributed storage systems, e.g. replication which is the traditional way. In this way, many copies of each piece of data fragment are contained to distinct storage nodes. As the amount of data increases, replication causes large storage overhead. Furthermore, erasure codes are introduced to decrease the repair cost. For instance, Reed-Solomon codes are utilized, however, the repair cost of them is not small. Therefore, locally recoverable/repairable codes have been developed to efficiently repair the node failures. These codes have attracted attention since they have theoretical advantages and applications in storage systems. In Chapter 1 of this thesis, development of locally recoverable codes is mentioned. In Chapter 2, basics of the algebraic coding theory are given. In addition, some definitions about graph theory that will be needed throughout the thesis are explained. In Chapter 3, some considerable definitions, bounds and theorems about locally recoverable codes in the literature are collected. Also, codes with locality and availability are mentioned. Some bounds of these codes are given. Codes with $(r,\delta)$-locality and some bounds of them are explained. Lastly, sequential-recovery LRCs and hiearchical LRCs are given. In Chapter 4, by using the incidence matrices of some special graphs, $(r,t)$-regular matrices are obtained and the parameters of locally recoverable codes are given. Moreover, locally recoverable codes are constructed by obtaining $(r,t)$-regular matrices from the cyclotomic cosets and magic squares. In Chapter 5, locality of constacyclic codes is described by means of their duals and two constructions are provided. Firstly, constacyclic codes with locality at most $2$ are found and a necessary and sufficient condition for constacyclic codes with locality $1$ is explained. By this construction, codes that more useful in practice since they have not only efficient encoding-decoding procedures thanks to being constacyclic but also locality $1$ are constructed. Further, this construction is generalized to constacyclic codes with any locality. Also, constacyclic LRCs that are optimal and almost optimal are examined. Moreover, optimal constacyclic LRCs whose distance is $2$ and constacyclic codes whose their locality is $1$ are obtained. Constacyclic codes with locality which is equal to their dimension are obtained. Lastly, constacyclic LRCs of length $n$ over $\mathbb{F}_q$ with condition $q\leq n$ are obtained by means of the cyclotomic cosets composed of elements that are relatively prime. In the last section the conclusions and future research directions are given.