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Register a new userCombinatorial Alphabet-Dependent Bounds for Locally Recoverable Codes
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Added by
Arya Mazumdar
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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Locally recoverable (LRC) codes have recently been a focus point of research in coding theory due to their theoretical appeal and applications in distributed storage systems. In an LRC code, any erased symbol of a codeword can be recovered by accessing only a small number of other symbols. For LRC codes over a small alphabet (such as binary), the optimal rate-distance trade-off is unknown. We present several new combinatorial bounds on LRC codes including the locality-aware sphere packing and Plotkin bounds. We also develop an approach to linear programming (LP) bounds on LRC codes. The resulting LP bound gives better estimates in examples than the other upper bounds known in the literature. Further, we provide the tightest known upper bound on the rate of linear LRC codes with a given relative distance, an improvement over the previous best known bounds.
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Motivated by applications in distributed storage, the notion of a locally recoverable code (LRC) was introduced a few years back. In an LRC, any coordinate of a codeword is recoverable by accessing only a small number of other coordinates. While different properties of LRCs have been well-studied, their performance on channels with random erasures or errors has been mostly unexplored. In this note, we analyze the performance of LRCs over such stochastic channels. In particular, for input-symmetric discrete memoryless channels, we give a tight characterization of the gap to Shannon capacity when LRCs are used over the channel.
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 than one erasure, which are subfield-subcodes of some $J$-affine variety codes. For these LRC codes, we compute localities $(r, delta)$ that determine the minimum size of a set $bar{R}$ of positions so that any $delta- 1$ erasures in $bar{R}$ can be recovered from the remaining $r$ coordinates in this set. We also show that some of these LRC codes with lengths $ngg q$ are $(delta-1)$-optimal.
We give a method to construct Locally Recoverable Error-Correcting codes. This method is based on the use of rational maps between affine spaces. The recovery of erasures is carried out by Lagrangian interpolation in general and simply by one addition in some good cases.
Recently, it was discovered by several authors that a $q$-ary optimal locally recoverable code, i.e., a locally recoverable code archiving the Singleton-type bound, can have length much bigger than $q+1$. This is quite different from the classical $q$-ary MDS codes where it is conjectured that the code length is upper bounded by $q+1$ (or $q+2$ for some special case). This discovery inspired some recent studies on length of an optimal locally recoverable code. It was shown in cite{LXY} that a $q$-ary optimal locally recoverable code is unbounded for $d=3,4$. Soon after, it was proved that a $q$-ary optimal locally recoverable code with distance $d$ and locality $r$ can have length $Omega_{d,r}(q^{1 + 1/lfloor(d-3)/2rfloor})$. Recently, an explicit construction of $q$-ary optimal locally recoverable codes for distance $d=5,6$ was given in cite{J18} and cite{BCGLP}. In this paper, we further investigate construction optimal locally recoverable codes along the line of using parity-check matrices. Inspired by classical Reed-Solomon codes and cite{J18}, we equip parity-check matrices with the Vandermond structure. It is turns out that a parity-check matrix with the Vandermond structure produces an optimal locally recoverable code must obey certain disjoint property for subsets of $mathbb{F}_q$. To our surprise, this disjoint condition is equivalent to a well-studied problem in extremal graph theory. With the help of extremal graph theory, we succeed to improve all of the best known results in cite{GXY} for $dgeq 7$. In addition, for $d=6$, we are able to remove the constraint required in cite{J18} that $q$ is even.
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. In this article we develop an algorithm that computes a recovery structure as concise posible for an arbitrary linear code $mathcal{C}$ and a recovery method that realizes it. This algorithm also provides the locality and the dual distance of $mathcal{C}$. Complexity issues are studied as well. Several examples are included.
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