New Constructions of Optimal Cyclic (r, δ) Locally Repairable Codes From Their Zeros

• Published in: IEEE transactions on information theory Vol. 67; no. 3; pp. 1596 - 1608
• Main Authors: Qiu, Jing, Zheng, Dabin, Fu, Fang-Wei
• Format: Journal Article
• Language: English
• Published: New York IEEE 01-03-2021; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Distributed storage; Generators; Indexes; Linear codes; locally repairable codes; Mathematics; maximum distance separable (MDS) codes; optimal cyclic LRCs; Reed-Solomon codes; Research and development; singleton-like bounds; Storage systems; Technological innovation
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2020.3043759

An (r, δ)-locally repairable code ((r, δ)-LRC for short) was introduced by Prakash et al. [14] for tolerating multiple failed nodes in distributed storage systems, which was a generalization of the concept of r-LRCs produced by Gopalan et al. [5]. An (r, δ)-LRC is said to be optimal if it achieves the Singleton-like bound. Recently, Chen et al. [2] generalized the construction of cyclic r-LRCs proposed by Tamo et al. [19], [20] and constructed several classes of optimal (r, δ)-LRCs of length n for n (q-1) or n (q+1), respectively in terms of a union of the set of zeros controlling the minimum distance and the set of zeros ensuring the locality. Following the work of [2], [3], this paper first characterizes (r, δ)-locality of a cyclic code via its zeros. Then we construct several classes of optimal cyclic (r, δ)-LRCs of length n for n (q - 1) or n (q+1), respectively from the product of two sets of zeros. Our constructions include all optimal cyclic (r, δ)-LRCs proposed in [2], [3], and our method seems more convenient to obtain optimal cyclic (r, δ)-LRCs with flexible parameters. Moreover, many optimal cyclic (r, δ)-LRCs of length n for n (q - 1) or n (q + 1), respectively with (r + δ - 1) n can be obtained from our method.