Local recovery properties of capacity achieving codes

Abstract

Abstract-A code is called locally recoverable or repairable if any symbol of a codeword can be recovered by reading only a small (constant) number of other symbols. The notion of local recoverability is important in the area of distributed storage where a most frequent error-event is a single storage node failure. A common objective is to repair the node by downloading data from as few other storage node as possible.In this paper we study the basic error-correcting properties of a locally recoverable code. We provide tight upper and lower bound on the local-recoverability of a code that achieves capacity of a symmetric channel. In particular it is shown that, if the coderate is less than the capacity then for the optimal codes, the maximum number of codeword symbols required to recover one lost symbol must scale as log 1 .

Cooperative local repair in distributed storage

“…Comparing with (7), we see that Partition codes are near-optimal in this case. Here note that, it was shown in [33] that for a random erasure channel, the Partition codes are asymptotically optimal in terms of achieving capacity.…”

An upper bound on the size of locally recoverable codes

“…In particular, in Sec. V, we will show that the family of Simplex codes (dual of Hamming codes) are optimal locally recoverable 1 Recent literature has extended the study of locally recoverable codes to include security constraints [21], notions of repair bandwidth [12], [13], multiple local repair alternatives [20], and probabilistic erasure models [16] (See [5] for a survey). In this work, we concern ourselves with the original notion of locality as described in [8], [18].…”

“…In the language used in [18], H(I) would denote the "entropy" associated with X I . Here, the above definition is appropriate since our modeling is adversarial, i.e., we do not presuppose any distribution on the messages or the codebook (see, [16] where such assumptions have been made). However, the behavior of the function H(.)…”

Average Repair Read Cost of Linear Repairable Code Ensembles

“…그러한 부호를 부분접속 복구 부호(Locally repairable code, LRC) 라 하는데, 이에 대한 간단하 며 실용적인 연구가 있었으며 [11,12] 부호 파라미터와 부분성에 연구가 진행되었다 [13,14] . 또한 저장 공간과 부분성에 대한 trade-off를 주는 피라미드 부호가 제안 되었으며 [15] , 자연적으로 부분접속 복구 성질을 갖는 파운틴 부호에 대한 near-MDS 성질의 연구가 발표되 었다 [16] .…”