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 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. For this aim, we use the polynomial interpolation over
$\mathbb {Z}_{p^{s}}$
proposed by Gopalan. Next, we present the parity-check matrices for another family of
$\left ({r,\delta }\right)$
-LRCs over
$\mathbb {Z}_{p^{s}}$
, and construct two instances of STB-optimal
$\left ({r,\delta }\right)$
-LRCs. To the best of our knowledge, this paper presents the first study on ring-based LRCs. The proposed LRCs over
$\mathbb {Z}_{p^{s}}$
exhibit certain design restrictions compared to LRCs over
$\mathbb {F}_{p^{s}}$
. 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 cost-efficiency 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 ring-based LRCs.