Related papers: Local Correctability of Expander Codes
An $[n,k]$ code $\mathcal{C}$ is said to be locally recoverable in the presence of a single erasure, and with locality parameter $r$, if each of the $n$ code symbols of $\mathcal{C}$ can be recovered by accessing at most $r$ other code…
A maximum distance separable (MDS) array code is composed of $m\times (k+r)$ arrays such that any $k$ out of $k+r$ columns suffice to retrieve all the information symbols. Expanded-Blaum-Roth (EBR) codes and Expanded-Independent-Parity…
We introduce graphical error-correcting codes, a new notion of error-correcting codes on $[q]^n$, where a code is a set of proper $q$-colorings of some fixed $n$-vertex graph $G$. We then say that a set of $M$ proper $q$-colorings of $G$…
Folded Reed-Solomon codes are an explicit family of codes that achieve the optimal trade-off between rate and error-correction capability: specifically, for any $\eps > 0$, the author and Rudra (2006,08) presented an $n^{O(1/\eps)}$ time…
In this work it is shown that locally repairable codes (LRCs) can be list-decoded efficiently beyond the Johnson radius for a large range of parameters by utilizing the local error-correction capabilities. The corresponding decoding radius…
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…
We initiate a study of locally decodable codes with randomized encoding. Standard locally decodable codes are error correcting codes with a deterministic encoding function and a randomized decoding function, such that any desired message…
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…
We present a novel iterative decoding algorithm for Reed-Muller (RM) codes, which takes advantage of a graph representation of the code. Vertices of the considered graph correspond to codewords, with two vertices being connected by an edge…
An $(\epsilon,\phi)$-expander decomposition of a graph $G=(V,E)$ is a clustering of the vertices $V=V_{1}\cup\cdots\cup V_{x}$ such that (1) each cluster $V_{i}$ induces subgraph with conductance at least $\phi$, and (2) the number of…
Tanner codes are graph-based linear codes whose parity-check matrices can be characterized by a bipartite graph $G$ together with a linear inner code $C_0$. Expander codes are Tanner codes whose defining bipartite graph $G$ has good…
Error-correcting codes that admit local decoding and correcting algorithms have been the focus of much recent research due to their numerous theoretical and practical applications. An important goal is to obtain the best possible tradeoffs…
An index code is said to be locally decodable if each receiver can decode its demand using its side information and by querying only a subset of the transmitted codeword symbols instead of observing the entire codeword. Local decodability…
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…
We study the problem of graph clustering where the goal is to partition a graph into clusters, i.e. disjoint subsets of vertices, such that each cluster is well connected internally while sparsely connected to the rest of the graph. In…
We describe a new class of list decodable codes based on Galois extensions of function fields and present a list decoding algorithm. These codes are obtained as a result of folding the set of rational places of a function field using…
It follows from the Marcus-Spielman-Srivastava proof of the Kadison-Singer conjecture that if $G=(V,E)$ is a $\Delta$-regular dense expander then there is an edge-induced subgraph $H=(V,E_H)$ of $G$ of constant maximum degree which is also…
Motivated by distributed storage applications, we investigate the degree to which capacity achieving encodings can be efficiently updated when a single information bit changes, and the degree to which such encodings can be efficiently…
We show that locally repairable codes (LRCs) can be list decoded efficiently beyond the Johnson radius for a large range of parameters by utilizing the local error correction capabilities. The new decoding radius is derived and the…
Consider a distributed coding for computing problem with constant decoding locality, i.e., with a vanishing error probability, any single sample of the function can be approximately recovered by probing only constant number of compressed…