Upper bounds on maximum lengths of Singleton-optimal locally repairable codes

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. In the recent years there has been a great amount of work on this topic. One of the main problems in this topic is to determine the largest length of a q-ary Singleton-optimal locally repairable code for given locality and minimum distance. Unlike classical MDS codes, the maximum length of Singleton? Optimal locally repairable codes are very sensitive to minimum distance and locality. Thus, it is more challenging and complicated to investigate the maximum length of Singleton-optimal locally repairable codes. In literature, there has been already some research on this problem. However, most of work is concerned with some specific parameter regime such as small minimum distance and locality, and rely on the constraint that (r + 1)|n and recovery sets are disjoint, where r is locality and n is the code length. In this paper we study the problem for large range of parameters including the case where minimum distance is proportional to length. In addition, we also derive some upper bounds on the maximum length of Singleton-optimal locally repairable codes with small minimum distance by removing this constraint. It turns out that even without the constraint we still get better upper bounds for codes with small locality and distance compared with known results. Furthermore, based on our upper bounds for codes with small distance and locality and some propagation rule that we propose in this paper, we are able to derive some upper bounds for codes with relatively large distance and locality assuming that (r + 1)|n and recovery sets are disjoint.