Related papers: Hierarchical Erasure Correction of Linear Codes
We present a parallel hierarchical solver for general sparse linear systems on distributed-memory machines. For large-scale problems, this fully algebraic algorithm is faster and more memory-efficient than sparse direct solvers because it…
Hypergraphs offer flexible and robust data representations for many applications, but methods that work directly on hypergraphs are not readily available and tend to be prohibitively expensive. Much of the current analysis of hypergraphs…
This thesis makes several significant contributions to the theory of both Regenerating (RG) and Locally Recoverable (LR) codes. The two principal contributions are characterizing the optimal rate of an LR code designed to recover from $t$…
Locally repairable codes (LRCs) are considered with equal or unequal localities, local distances and local field sizes. An explicit two-layer architecture with a sum-rank outer code is obtained, having disjoint local groups and achieving…
Erasure codes are an integral part of many distributed storage systems aimed at Big Data, since they provide high fault-tolerance for low overheads. However, traditional erasure codes are inefficient on reading stored data in degraded…
Using linear functional-based duality of modules, we generalize the syndrome decoding algorithm of linear codes over finite fields to those over finite commutative rings. Moreover, If the ring is local the algorithm is simplified by…
Erasure codes provide a storage efficient alternative to replication based redundancy in (networked) storage systems. They however entail high communication overhead for maintenance, when some of the encoded fragments are lost and need to…
Recently, Martinez-Penas and Kschischang (IEEE Trans. Inf. Theory, 2019) showed that lifted linearized Reed-Solomon codes are suitable codes for error control in multishot network coding. We show how to construct and decode lifted…
Ensemble models are widely used to solve complex tasks by their decomposition into multiple simpler tasks, each one solved locally by a single member of the ensemble. Decoding of error-correction codes is a hard problem due to the curse of…
A linear error correcting code is a subspace of a finite-dimensional space over a finite field with a fixed coordinate system. Such a code is said to be locally recoverable with locality $r$ if, for every coordinate, its value at a codeword…
We introduce a new family of erasure codes, called group decodable code (GDC), for distributed storage system. Given a set of design parameters {\alpha; \beta; k; t}, where k is the number of information symbols, each codeword of an…
This paper investigates the design and application of write-once memory (WOM) codes for flash memory storage. Using ideas from Merkx ('84), we present a construction of WOM codes based on finite Euclidean geometries over $\mathbb{F}_2$.…
Erasure-correcting codes, that support local repair of codeword symbols, have attracted substantial attention recently for their application in distributed storage systems. This paper investigates a generalization of the usual locally…
In this paper we investigate the behavior of iteratively decoded low-density parity-check codes over the binary erasure channel in the so-called ``waterfall region." We show that the performance curves in this region follow a very basic…
Erasure list decoding was introduced to correct a larger number of erasures with output of a list of possible candidates. In the present paper, we consider both random linear codes and algebraic geometry codes for list decoding erasure…
A generic construction of linear codes over finite fields has recently received a lot of attention, and many one-weight, two-weight and three-weight codes with good error correcting capability have been produced with this generic approach.…
A simple algorithm for decoding both errors and erasures of Reed-Solomon codes is described.
We study the symmetric tensor rank of multiplication over finite field extensions using linearized polynomials. Via field trace, symmetric linearized polynomials are identified with symmetric bilinear forms and symmetric matrices, allowing…
Linear codes are widely employed in communication systems, consumer electronics, and storage devices. All linear codes over finite fields can be generated by a generator matrix. Due to this, the generator matrix approach is called a…
While quantum algorithms for solving large scale systems of linear equations offer potentially exponential speedups, their application has largely been confined to sparse matrices. This work extends the scope of these algorithms to a broad…