Bounded-degree Low Rank Parity Check Codes

Abstract: Low-rank parity-check (LRPC) codes are the rank-metric analogue of low-density parity-check codes and they found important applications in code-based cryptography. In this paper we investigate a sub-family of LRPC codes, which have a parity-check matrix defined over a subspace $\calV_{\alpha,d}=\Span{\Fq}{1,\alpha, \ldots, \alpha^{d-1}}\subsetneq \Fqm$, where $\Fqm$ is the finite field of $q^m$ elements and $d$ is a positive integer significantly smaller than $m $; and they are termed bounded-degree LRPC (BD-LRPC) codes. These codes are the same as the standard LRPC codes of density $2$ when the degree $d=2$, while for degree $d>2$ they constitute a proper subset of LRPC codes of density $d$. Exploiting the structure of $\calV_{\alpha,d}$, 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. This underscores the superior decoding capability of the BD-LRPC codes. Moreover, as the code length $n\rightarrow \infty$, when $n/m\rightarrow 0$, the BD-LRPC codes with a code rate of $R=k/n$ can be uniquely decodable with radius $\rho=r/n$ approaching the Singleton bound $1-R$ by letting $\epsilon=u/n\rightarrow 0$; and when $n/m$ is a constant, the BD-LRPC codes can have unique decoding radius $\rho = 1-R-\epsilon $ for a small $\epsilon$, allowing for $\rho>(1-R)/2$ with properly chosen parameters. This superior decoding capability is theoretically proved for the case $d=2$ and confirmed by experimental results for $d>2$.