Related papers: List Decoding of Locally Repairable Codes
In this paper, we show that the projective Reed-Muller~(PRM) codes form a family of locally correctable codes~(LCC) in the regime of low query complexities. A PRM code is specified by the alphabet size $q$, the number of variables $m$, and…
The sum-rank metric is the mixture of the Hamming and rank metrics. The sum-rank metric found its application in network coding, locally repairable codes, space-time coding, and quantum-resistant cryptography. Linearized Reed-Solomon (LRS)…
We construct the first (locally computable, approximately) locally list decodable codes with rate, efficiency, and error tolerance approaching the information theoretic limit, a core regime of interest for the complexity theoretic task of…
We study the Singleton-type bound that provides an upper limit on the minimum distance of locally repairable codes. We present an improved bound by carefully analyzing the combinatorial structure of the repair sets. Thus, we show the…
A major issue of locally repairable codes is their robustness. If a local repair group is not able to perform the repair process, this will result in increasing the repair cost. Therefore, it is critical for a locally repairable code to…
Error-correcting codes are one of the most fundamental objects in pseudorandomness, with applications in communication, complexity theory, and beyond. Codes are useful because of their ability to support decoding, which is the task of…
Locally recoverable codes (LRCs) with locality parameter $r$ can recover any erased code symbol by accessing $r$ other code symbols. This local recovery property is of great interest in large-scale distributed classical data storage systems…
A Locally Recoverable code is an error-correcting code such that any erasure in a single coordinate of a codeword can be recovered from a small subset of other coordinates. We study Locally Recoverable Algebraic Geometry codes arising from…
Reed-Solomon codes are a classic family of error-correcting codes consisting of evaluations of low-degree polynomials over a finite field on some sequence of distinct field elements. They are widely known for their optimal unique-decoding…
Algebraic decoding algorithms are commonly applied for the decoding of Reed-Solomon codes. Their main advantages are low computational complexity and predictable decoding capabilities. Many algorithms can be extended for correction of both…
We give a new framework based on graph regularity lemmas, for list decoding and list recovery of codes based on spectral expanders. Using existing algorithms for computing regularity decompositions of sparse graphs in (randomized)…
In this paper, the repair problem for erasures beyond locality in locally repairable codes is explored under a practical system setting, where a rack-aware storage system consists of racks, each containing a few parity checks. This is…
A k-query Locally Decodable Code (LDC) encodes an n-bit message x as an N-bit codeword C(x), such that one can probabilistically recover any bit x_i of the message by querying only k bits of the codeword C(x), even after some constant…
Folded Reed-Solomon (FRS) codes are a well-studied family of codes, known for achieving list decoding capacity. In this work, we give improved deterministic and randomized algorithms for list decoding FRS codes of rate $R$ up to radius…
An $(n,r,h,a,q)$-Local Reconstruction Code (LRC) is a linear code over $\mathbb{F}_q$ of length $n$, whose codeword symbols are partitioned into $n/r$ local groups each of size $r$. Each local group satisfies `$a$' local parity checks to…
List-decoding and list-recovery are important generalizations of unique decoding that received considerable attention over the years. However, the optimal trade-off among list-decoding (resp. list-recovery) radius, list size, and the code…
A $q$-ary $(n,k,r)$ locally repairable code (LRC) is an $[n,k,d]$ linear code over $\mathbb{F}_q$ such that every code symbol can be recovered by accessing at most $r$ other code symbols. The well-known Singleton-like bound says that $d \le…
The matrix completion problem provides a unifying lens through which many fundamental problems in coding theory can be viewed. In this paper, we investigate Locally Recoverable Codes (LRCs) with Maximal Recoverability (MR) and Maximum…
In this work, we prove new results concerning the combinatorial properties of random linear codes. Firstly, we prove a lower bound on the list-size required for random linear codes over $\mathbb F_q$ $\varepsilon$-close to capacity to…
Error-correcting codes are a method for representing data, so that one can recover the original information even if some parts of it were corrupted. The basic idea, which dates back to the revolutionary work of Shannon and Hamming about a…