On the Locality of Codeword Symbols

Consider a linear [n, k, d]q code C. We say that that i-th 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 the locality of parity check symbols and erasure correction beyond worst case distance for optimal codes. Usin...

• Type: Journal
• Year: 2011
• Where: CORR
• Authors: Parikshit Gopalan, Cheng Huang, Huseyin Simitci, Sergey Yekhanin
• Keywords: CORR 2011, Education, Low Locality, Parity Checks, Tight Bound