  Low rank parity check (LRPC) codes are the rank-metric analogue of low
density parity check codes. In this paper we investigate a sub-family of LRPC
codes, which have a parity check matrix defined over a subspace
$V_{\alpha,d}=\langle 1,\alpha, \ldots,
\alpha^{d-1}\rangle_{\mathbb{F}_q}\subsetneq \mathbb{F}_{q^m}$, where
$\mathbb{F}_{q^m}$ is the finite field of $q^m$ elements and $d$ is
significantly smaller than $m $. These codes are named bounded-degree LRPC
(BD-LRPC) codes and are the same as the standard LRPC codes of density $2$ when
the degree $d=2$, while BD-LRPC codes of degree $d>2$ constitute a proper
subset of LRPC codes of density $d$. Exploiting the particular structure of
their parity check matrix, we show that the BD-LRPC codes of degree $d$ can
uniquely correct errors of rank weight $r$ when $n-k \geq r + u$ for certain $u
\geq 1$, in contrast to the condition $n-k\geq dr$ required for the standard
LRPC codes, where $d\geq n/(n-k)$. This underscores the superior decoding
capability of the proposed BD-LRPC codes. As the code length $n$ approaches
infinity, when $n/m\rightarrow 0$, it is shown that $u$ can be chosen as a
certain constant, which indicates that the BD-LRPC codes with a code rate of
$R$ can be, with a high probability, uniquely decodable with the decoding
radius $\rho=r/n$ approaching the Singleton bound $1-R$ for $n \to \infty$; and
when $b= n/m$ is a constant, the BD-LRPC codes can have unique decoding radius
$\rho = 1-R-\epsilon $ for a small $\epsilon$, which can easily lead to
$\rho>(1-R)/2$ with properly chosen parameters.
