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Paper IPM / M / 17578  


Abstract:  
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 $\delta1$ nodes by
accessing information from $r$ other nodes. In this paper, we present new constructions of $\left(r,\delta\right)$LRCs, with $2\leq\delta \leq \frac{p1}{t}$ over $\mathbb{Z}_{p^s}$, where $t$ divides $p1$ and $t\neq p1$. Initially, we provide generator matrices for $\left(r,2\right)$LRCs, among which one instance is considered as SingletonType Bound (STB)optimal, a notion introduced in this paper. Also, we present a method for recovering an erased symbol in a codeword of our $\left(r,2\right)$LRC. \textcolor{mycolor2}{For this aim, we use the polynomial interpolation over $\mathbb{Z}_{p^s}$ proposed by Gopalan. Next, we present the paritycheck matrices for another family of $\left(r,\delta\right)$LRCs over $\mathbb{Z}_{p^s}$, and construct two instances of STBoptimal $\left(r,\delta\right)$LRCs.
To the best of our knowledge, this paper presents the first study on ringbased LRCs. The proposed LRCs over $\mathbb{Z}_{p^s}$ exhibit certain design restrictions compared to LRCs over $\mathbb{F}_{p^s}$. \textcolor{mycolor2_rev2}{However, we provide two advantages for LRCs over $\mathbb{Z}_{p^s}$. First, we analyze Boolean circuits for arithmetic operations and demonstrate that the complexity of implementing multiplication in $\mathbb{Z}_{p^s}$, the operation with the highest cost in our algorithms, is considerably lower than in $\mathbb{F}_{p^s}$.
This highlights the superior performance of our LRCs in terms of implementation speed and costefficiency compared to their counterparts. Next, we offer an example illustrating that the Gray image of particular $\mathbb{Z}_{p^{s+1}}$LRCs of length $n$ results in LRCs of length $np^s$ over $\mathbb{F}_{p}$, 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 ringbased LRCs.
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