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 (r, δ)-LRC if for each i, the ith component of codewords have locality (r, δ), that is, there exists a punctured subcode of C with support containing i, whose length is at most r + δ-1, and whose minimum distance is at least δ. An (r, δ)-LRC with locality (r, δ) allows for the local recovery of any δ-1 nodes by accessing information from r other nodes. In this paper, we present new constructions of (r, δ)-LRCs, with 2 ≤ δ ≤ p-1/t over Z _p^s , where t divides p-1 and t ≠ p - 1. Initially, we provide generator matrices for (r, 2)-LRCs, among which one instance is considered as Singleton-Type Bound (STB)-optimal, a notion introduced in this paper. Also, we present a method for recovering an erased symbol in a codeword of our (r, 2)-LRC. For this aim, we use the polynomial interpolation over Z _p^s proposed by Gopalan. Next, we present the parity-check matrices for another family of (r, δ)-LRCs over Z _p^s , and construct two instances of STB-optimal (r, δ)-LRCs. To the best of our knowledge, this paper presents the first study on ring-based LRCs. The proposed LRCs over Z _p^s exhibit certain design restrictions compared to LRCs over F _p^s . However, we provide two advantages for LRCs over Z _p^s . First, we analyze Boolean circuits for arithmetic operations and demonstrate that the complexity of implementing multiplication in Z _p^s , the operation with the highest cost in our algorithms, is considerably lower than in F _p^s . This highlights the superior performance of our LRCs in terms of implementation speed and cost-efficiency compared to their counterparts. Next, we offer an example illustrating that the Gray image of particular Z _p^s+1 -LRCs of length n results in LRCs of length np^s over F _p , which may not necessarily be linear. This introduces a novel class of LRCs, prompting further exploration of the connections between existing nonlinear LRCs in finite fields and linear ring-based LRCs.