On Lengths of Singleton-Optimal Locally Repairable Codes
Shu Liu, Ting-Yi Wu, Chaoping Xing, Chen Yuan · IEEE Transactions on Communications · 2023
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 aq-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)|nand the recovery sets are disjoint so as to simplify the argument,whereris locality andnis 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)|nand the recovery sets are disjoint.It 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)|nand recovery sets are disjoint.