On the Locality of Codeword Symbols

Abstract

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. Using our structure theorem, we obtain a tight bound for the locality of parity symbols possible in such codes for a broad class of parameter settings. We prove that there is a tradeoff between having good locality for parity checks and the ability to correct erasures beyond the minimum distance.

Generalized construction of cyclic (<i>r</i>, <i>t</i>)-locally repairable codes using trace function

“…t þ 1, note that there are t ¼ eðm À þ 1Þ polynomials that contain the common term x q À1 as shown in (1). Each of these polynomials corresponds to a row in a parity check matrix of a one-step majority-logic decodable code [4].…”

“…Locally repairable codes (LRCs) [1,2] have recently attracted considerable attention as a promising technology for data protection. An ðr; tÞ-LRC is a code having the following property: each coordinate of codewords can be recovered from at most r other coordinates (called a repair set), and there are at least t disjoint repair sets for each coordinate, where r and t are called locality and availability, respectively [3].…”

Binary Locally Repairable Codes from Complete Multipartite Graphs

“…분산 저장 부호화 방법의 성능을 평가하는 중요한 지표 중 하나인 부분접속수(locality) [2] 는 임의의 손실 된 노드를 복구하기 위하여 접속해야 하는 최소 노드 의 수를 의미한다. 부분접속수의 이론적 한계식 [2] 이 제시된 이후 부분접속 복구 부호에 대한 연구와 좀 더 일반화된 부분접속수에 대한 연구가 활발히 진행되고 있다 [3][4][5][6][7][8][9][10][11] .…”

“…부분접속수의 이론적 한계식 [2] 이 제시된 이후 부분접속 복구 부호에 대한 연구와 좀 더 일반화된 부분접속수에 대한 연구가 활발히 진행되고 있다 [3][4][5][6][7][8][9][10][11] . 특히 최근 임의의 노드 개가 손실된 경우 손실된 개의 노드들을 복구하기 위하여 이용되어야 하는 최소 노드의 개수를 -부분접속수(  )로 정의하 고, -부분접속수를 만족하는 부호 설계에 대한 연구 가 발표되었다 [11] .…”

Local recovery properties of capacity achieving codes

“…This property is desirable in distributed storage systems and was introduced in that context in [7].…”