Optimal Linear and Cyclic Locally Repairable Codes over Small Fields

We consider locally repairable codes over small fields and propose constructions of optimal cyclic and linear codes in terms of the dimension for a given distance and length. Four new constructions of optimal linear codes over small fields with locality properties are developed. The first two approaches give binary cyclic codes with locality two. While the first construction has availability one, the second binary code is characterized by multiple available repair sets based on a binary Simplex code. The third approach extends the first one to q-ary cyclic codes including (binary) extension fields, where the locality property is determined by the properties of a shortened first-order Reed-Muller code. Non-cyclic optimal binary linear codes with locality greater than two are obtained by the fourth construction.Comment: IEEE Information Theory Workshop (ITW) 2015, Apr 2015, Jerusalem, Israe

Capacity of Locally Recoverable Codes

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.Comment: Invited paper to the Information Theory Workshop (ITW) 201

Combinatorial Alphabet-Dependent Bounds for Locally Recoverable Codes

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.Comment: To appear in IEEE Transactions on Information Theor

Constructions of Binary Optimal Locally Repairable Codes via Intersection Subspaces

Locally repairable codes (LRCs), which can recover any symbol of a codeword by reading only a small number of other symbols, have been widely used in real-world distributed storage systems, such as Microsoft Azure Storage and Ceph Storage Cluster. Since binary linear LRCs can significantly reduce coding and decoding complexity, constructions of binary LRCs are of particular interest. The aim of this paper is to construct dimensional optimal binary locally repairable codes with disjoint local repair groups. We introduce how to connect intersection subspaces with binary locally repairable codes and construct dimensional optimal binary linear LRCs with locality $2^b$ ($b\geq 3$) and minimum distance $d\geq 6$ by employing intersection subspaces deduced from the direct sum. This method will sufficiently increase the number of possible repair groups of dimensional optimal LRCs, and thus efficiently expanding the range of the construction parameters while keeping the largest code rates compared with all known binary linear LRCs with minimum distance $d\geq 6$ and locality $2^b$ ($b\geq 3$).Comment: Accepted for publication in the SCIENCE CHINA Information Science

Malleable coding for updatable cloud caching

In software-as-a-service applications provisioned through cloud computing, locally cached data are often modified with updates from new versions. In some cases, with each edit, one may want to preserve both the original and new versions. In this paper, we focus on cases in which only the latest version must be preserved. Furthermore, it is desirable for the data to not only be compressed but to also be easily modified during updates, since representing information and modifying the representation both incur cost. We examine whether it is possible to have both compression efficiency and ease of alteration, in order to promote codeword reuse. In other words, we study the feasibility of a malleable and efficient coding scheme. The tradeoff between compression efficiency and malleability cost-the difficulty of synchronizing compressed versions-is measured as the length of a reused prefix portion. The region of achievable rates and malleability is found. Drawing from prior work on common information problems, we show that efficient data compression may not be the best engineering design principle when storing software-as-a-service data. In the general case, goals of efficiency and malleability are fundamentally in conflict.This work was supported in part by an NSF Graduate Research Fellowship (LRV), Grant CCR-0325774, and Grant CCF-0729069. This work was presented at the 2011 IEEE International Symposium on Information Theory [1] and the 2014 IEEE International Conference on Cloud Engineering [2]. The associate editor coordinating the review of this paper and approving it for publication was R. Thobaben. (CCR-0325774 - NSF Graduate Research Fellowship; CCF-0729069 - NSF Graduate Research Fellowship)Accepted manuscrip