Complete Characterizations of Optimal Locally Repairable Codes With Locality 1 and <inline-formula> <tex-math notation="LaTeX">$K-1$ </tex-math></inline-formula>

• Main Authors: Yichong Xia, Bin Chen
• Format: Article
• Language: English
• Published: IEEE 2019-01-01
• Series: IEEE Access
• Subjects: Distributed storage systems; erasure codes; maximum distance separable (MDS) codes; near MDS codes; locally repairable codes
• Online Access: https://ieeexplore.ieee.org/document/8794791/

A locally repairable code (LRC) is a [n, k, d] linear code with length n, dimension k, minimum distance d and locality r, which means that every code symbol can be repaired by at most r other symbols. LRCs have become an important candidate in distributed storage systems due to their relatively low I/O cost. An LRC is said to be optimal if its minimum distance meets one of the Singleton-like bounds. This paper considers the optimal constructions of LRCs with locality r = 1 and r = k - 1, which involves three types: r-local LRCs, (r, &#x03B4;)-LRCs and LRCs with t-availability. Specifically, we first prove that the existence of an optimal LRC with locality r = 1 is equivalent to that of an MDS code with certain parameters. Thus we can completely characterize the three types of optimal LRCs with r = 1 based on some known constructions of MDS codes. Near MDS codes is a special class of sub-optimal linear code whose minimum distance d = n - k and the i-th generalized Hamming weight achieves the generalized Singleton bound for 2 &#x2264; i &#x2264; k. For r = k - 1, we have established the connections between optimal r-local LRCs/ LRCs with t-availability and near MDS codes. Such connections can help to construct optimal LRCs with r = k - 1 from some known classes of near MDS codes.