Related papers: Maximal Recoverability: A Nexus of Coding Theory
Classical locally recoverable codes (LRCs) have become indispensable in distributed storage systems. They provide efficient recovery in terms of localized errors. Quantum LRCs have very recently been introduced for their potential…
A code over a finite field is called locally recoverable code (LRC) if every coordinate symbol can be determined by a small number (at most r, this parameter is called locality) of other coordinate symbols. For a linear code with length n,…
Regenerating codes are efficient methods for distributed storage in storage networks, where node failures are common. They guarantee low cost data reconstruction and repair through accessing only a predefined number of arbitrarily chosen…
We establish a connection between problems studied in rigidity theory and matroids arising from linear algebraic constructions like tensor products and symmetric products. A special case of this correspondence identifies the problem of…
Regenerating codes are a class of codes proposed for providing reliability of data and efficient repair of failed nodes in distributed storage systems. In this paper, we address the fundamental problem of handling errors and erasures during…
Fractional repetition (FR) codes are a family of repair-efficient storage codes that provide exact and uncoded node repair at the minimum bandwidth regenerating point. The advantageous repair properties are achieved by a tailor-made…
The maximal coding rate reduction (MCR$^2$) objective for learning structured and compact deep representations is drawing increasing attention, especially after its recent usage in the derivation of fully explainable and highly effective…
Designing good error correcting codes whose generator matrix has a support constraint, i.e., one for which only certain entries of the generator matrix are allowed to be non-zero, has found many recent applications, including in distributed…
Partial maximum distance separable (PMDS) codes are a kind of erasure codes where the nodes are divided into multiple groups with each forming an MDS code with a smaller code length, thus they allow repairing a failed node with only a few…
A code is said to be a Locally Recoverable Code (LRC) with availability if every coordinate can be recovered from multiple disjoint sets of other coordinates called recovering sets. The vector of sizes of recovering sets of a coordinate is…
We investigate the distance properties of linear locally recoverable codes (LRC codes) with all-symbol locality and availability. New upper and lower bounds on the minimum distance of such codes are derived. The upper bound is based on the…
The locally repairable code (LRC) studied in this paper is an $[n,k]$ linear code of which the value at each coordinate can be recovered by a linear combination of at most $r$ other coordinates. The central problem in this work is to…
List recovery of error-correcting codes has emerged as a fundamental notion with broad applications across coding theory and theoretical computer science. Folded Reed-Solomon (FRS) and univariate multiplicity codes are explicit…
Maximum distance separable (MDS) are constructed to required specifications. The codes are explicitly given over finite fields with efficient encoding and decoding algorithms. Series of such codes over finite fields with ratio of distance…
A code is locally recoverable when each symbol in one of its code words can be reconstructed as a function of $r$ other symbols. We use bundles of projective spaces over a line to construct locally recoverable codes with availability; that…
The continuously increasing amount of digital data generated by today's society asks for better storage solutions. This survey looks at a new generation of coding techniques designed specifically for the needs of distributed networked…
Locally recoverable codes (LRCs) have emerged as fundamental objects in modern coding theory, primarily due to their pivotal role in distributed and cloud storage systems. A major breakthrough in their construction was achieved by Tamo and…
A locally recoverable (LRC) code is a code over a finite field $\mathbb{F}_q$ such that any erased coordinate of a codeword can be recovered from a small number of other coordinates in that codeword. We construct LRC codes correcting more…
We consider error decoding of locally repairable codes (LRC) and partial MDS (PMDS) codes through interleaved decoding. For a specific class of LRCs we investigate the success probability of interleaved decoding. For PMDS codes we show that…
We consider linear network error correction (LNEC) coding when errors may occur on edges of a communication network of which the topology is known. In this paper, we first revisit and explore the framework of LNEC coding, and then unify two…