A classification for optimal quaternary locally repairable codes
Xi Yuanxiao, Kong Xiangliang, Gennian Ge · Scientia Sinica Mathematica · 2022
In recent years, several new types of codes have been introduced to provide the fault-tolerance and guarantee the system reliability in distributed storage systems, among which locally repairable codes (LRCs) have played an important role. A linear code is said to have locality $r$ if each of its code symbols can be repaired by accessing at most $r$ other code symbols. For an LRC with length $n$, dimension $k$, and locality $r$, its minimum distance $d$ has been proved to satisfy the Singleton-like bound $d\leq~n-k-\lceil~k/r\rceil+2$. Since then, many studies have been done for constructing LRCs meeting the Singleton-like bound over small fields. In this paper, we study quaternary LRCs meeting the Singleton-like bound through a parity-check matrix approach, using tools from combinatorial designs and finite geometry. We prove that there are $27$ different classes of parameters for optimal quaternary LRCs. Moreover, for each class, explicit constructions of corresponding optimal quaternary LRCs are presented. In addition, using tools from finite geometry, we also introduce a new method to determine the existence of the optimal LRCs.