On Lengths of Singleton-Optimal Locally Repairable Codes

作     者：Liu, Shu Wu, Ting-Yi Xing, Chaoping Yuan, Chen 

作者机构：Univ Elect Sci & Technol China Natl Key Lab Wireless Commun Chengdu 611731 Peoples R China Huawei Technol Co Ltd Cent Res Inst Theory Lab Labs 2012 Hong Kong 999077 Peoples R China Shanghai Jiao Tong Univ Sch Elect Informat & Elect Engn Shanghai 200240 Peoples R China 

出 版 物：《IEEE TRANSACTIONS ON COMMUNICATIONS》 (IEEE Trans Commun)

年 卷 期：2024年第72卷第1期

页      面：3-12页

核心收录：

学科分类：0810[工学-信息与通信工程] 0808[工学-电气工程] 08[工学] 

基　　金：National Key Research and Development Program of China 

主　　题：Codes Upper bound Symbols Reed-Solomon codes Encoding Distributed databases Codecs Locally repairable codes singleton-optimal upper bounds minimum distance 

摘      要：A locally repairable code is called Singleton-optimal if it achieves the Singleton-type bound. Such codes are of great theoretic interest in the study of locally repairable codes. One of the major problems in this topic is to determine the maximum length of a q -ary Singleton-optimal locally repairable code with fixed locality and minimum distance. Unlike classical MDS codes, the maximum length of Singleton-optimal locally repairable codes is very sensitive to the minimum distance and locality. Thus, determining the maximum length of the Singleton-optimal locally repairable codes is more challenging and complicated. In literature, many efforts are paid to solve this problem especially for small distance and locality regime. Moreover, most of works also requires that (r+1)|n and the recovery sets are disjoint so as to simplify the argument, where r is locality and n is the code length. In this paper, we derive some upper bounds on the maximum length of Singleton-optimal locally repairable codes with minimum distance 5,6 and 7 without the constraint that (r+1)|n and the recovery sets are *** turns out that even without this constraint we still obtain better upper bounds for codes with small locality and distance compared to known results. Furthermore, based on our upper bounds for codes with small distance and locality, we propose the propagation rule to derive some upper bounds for codes with relatively large distance and locality assuming that (r+1)|n and recovery sets are disjoint.