Locally Recoverable Codes Over Zp s

Locally recoverable codes (LRCs) play a vital role in distributed storage systems where the failure or unavailability of storage devices is a common occurrence. The purpose of LRCs is to facilitate the repair processes required to recover lost or damaged data in such systems. A code C will be said \left ({r,\delta }\right) -LRC if for each i , the i th component of codewords have locality (r, \delta) , that is, there exists a punctured subcode of C with support containing i , whose length is at most r + \delta - 1 , and whose minimum distance is at least \delta . An (r, \delta) -LRC with locality (r, \delta) allows for the local recovery of any \delta -1 nodes by accessing information from r other nodes. In this paper, we present new constructions of \left ({r,\delta }\right) -LRCs, with 2\leq \delta \leq \frac {p-1}{t} over \mathbb {Z}_{p^{s}} , where t divides p-1 and t\neq p-1 . Initially, we provide generator matrices for \left ({r,2}\right) -LRCs, among which one instance is considered as Singleton-Type Bound (STB)-optimal, a notio