On the Locality of Codeword Symbols

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On the Locality of Codeword Symbols

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Creation Date: Jul 06, 2014

Published In: Nov 2012

Paper Type: Journal Article

Book Title: IEEE Transactions on Information Theory

Abstract:

Consider a linear [n,k,d]q code C. We say that the ith coordinate of C has locality r , if the value at this coordinate can be recovered from accessing some other r coordinates of C. Data storage applications require codes with small redundancy, low locality for information coordinates, large distance, and low locality for parity coordinates. In this paper, we carry out an in-depth study of the relations between these parameters. We establish a tight bound for the redundancy n-k in terms of the message length, the distance, and the locality of information coordinates. We refer to codes attaining the bound as optimal. We prove some structure theorems about optimal codes, which are particularly strong for small distances. This gives a fairly complete picture of the tradeoffs between codewords length, worst case distance, and locality of information symbols. We then consider thelocality of parity check symbols and erasure correction beyond worst case distance for optimal codes. Using our structure theorem, we obtain a tight bound for the locality of parity symbolspossible in such codes for a broad class of parameter settings. We prove that there is a tradeoff between having good locality and the ability to correct erasures beyond the minimum distance.

IEEE Explore link: http://dx.doi.org/10.1109/TIT.2012.2208937

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