Bounded-degree Low Rank Parity Check Codes
Ermes Franch, Chunlei Li
TL;DR
This work introduces bounded-degree LRPC (BD-LRPC) codes, a subfamily of low-rank parity-check codes whose parity-check support lies in the subspace $\\mathcal{V}_{\\alpha,d}$. It develops a decoding strategy based on expanding the syndrome subspace via $\\mathcal{V}_{\\alpha,t}$ to align $\\mathcal{V}_{\\alpha,t}.\\mathcal{S}$ with $\\mathcal{V}_{\\alpha,d+t-1}.\\mathcal{E}$, enabling recovery of the error support and the original error under $d(n-k)\\ge nr$. The paper proves unique decodability with high probability for $d=2$ and provides experimental evidence for $d>2$, plus a conjecture to extend to general $d>2$ and $r>1$, showing the radius can approach $\rho \approx 1-R$ (and exceed $(1-R)/2$ for fixed $b=n/m$). This work offers a probabilistic solution to an open problem of Xing and Yuan and advances cryptographic applications by delivering BD-LRPC codes with decoding radii near the Singleton bound and structured subspace properties.
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 $\calV_{α,d}=\Span{\Fq}{1,α, \ldots, α^{d-1}}\subsetneq \Fqm$, 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 $\calV_{α,d}$, 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 $ρ=r/n$ approaching the Singleton bound $1-R$ by letting $ε=u/n\rightarrow 0$; and when $n/m$ is a constant, the BD-LRPC codes can have unique decoding radius $ρ= 1-R-ε$ for a small $ε$, allowing for $ρ>(1-R)/2$ with properly chosen parameters. This superior decoding capability is theoretically proved for the case $d=2$ and confirmed by experimental results for $d>2$.