Research Projects

Maximum distance separable (MDS) codes are optimal codes that attain the well-known Singleton bound. These codes achieve the highest possible Hamming distance for a given code length and size. As the Hamming distance of a code measures its error-detecting and error-correcting capabilities, these codes exhibit the maximum error-detecting and error-correcting capabilities for a given code length and size. On the other hand, additive codes have nice algebraic structures and are useful in constructing quantum stabilizer codes. In our recent work, we identified some new classes of additive codes that are MDS within the families of additive generalized Reed-Solomon (additive GRS) codes and additive generalized twisted Reed-Solomon (additive GTRS) codes. In another direction, distributed storage systems are increasingly being adopted for the storage of data, which is increasing at an exponential rate. In such systems, data is distributed across multiple nodes. One recurring problem in these systems is that of node failures. When a node fails, the data it contains is lost. To ensure reliability, it is essential to recover this lost data. To address this problem, coding schemes are increasingly being adopted. To that end, one approach to recover the lost data is to access the information stored in other neighboring nodes and retrieve the lost data using this information. This challenge is known as the repair problem. This raises a fundamental question: Can we design codes that allow efficient data recovery by accessing a smaller number of nodes while maintaining the same level of redundancy? The goal is to develop codes that enable quick reconstruction of lost data while preserving storage efficiency. In an effort to address this challenge, Gopalan et al. (2012) introduced the concept of locality in coding theory. The locality of a node is the smallest positive integer r such that the data stored on that node can be recovered by accessing at most r other nodes. This led to the development of locally recoverable codes (LRCs) with locality r. The main objectives of this project are: (1) to identify new classes of MDS codes within the families of additive GRS and additive GTRS codes. (2) To construct additive MDS self-orthogonal and self-dual codes w.r.t the Hermitian trace bi- linear form through additive GRS and GTRS codes. (3) To study and provide methods to construct MDS ACD codes over finite fields. (4) To study the locality properties of additive MDS codes over finite fields and identify optimal LRCs within this class. To develop efficient recovery methods for these codes, enhancing their practical applicability in distributed storage systems. (5) To explore quantum LRCs (qLRCS) by characterizing good polynomials over finite fields and constructing qLRCs from additive codes.