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${\mathcal {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}$elements,$\alpha \in \mathbb {F}_{q^{m}}$is an element not in any proper subfield of$\mathbb {F}_{q^{m}}$, and d is a positive integer significantly smaller than m. These codes are termed bounded-degree LRPC (BD-LRPC) codes. BD-LRPC codes are the same as the standard LRPC codes of density 2 when the degree$d=2$, while for degree$d\gt 2$they constitute a proper subset of LRPC codes of density d. Exploiting the structure of${\mathcal {V}}_{\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 \gt (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\gt 2$.

Let Fqbe the finite field withqelements andmbe a positive integer. The Fqm-linear low-rank parity-check (LRPC) codes have been used in many cryptographic schemes. Motivated by recent attacks on those schemes, this paper generalizes LRPC codes based on 3-tensors in Fm×m×mq. The generalized LRPC codes are mostly Fq-linear matrix codes, while a particular choice of the 3-tensor is isomorphic to the original Fqm-linear LRPC codes. We first introduce a bilinearT-product over Fmqassociated with a 3-tensorT∈ Fm×m×mq. Based on theT-product, we propose a generic method to expand Fq-linear matrix code from dimensionkto dimensionkmand then use the method to generalize LRPC codes. Finally, we propose two probabilistic polynomial-time decoding algorithms for the generalized LRPC codes under different circumstances. We provide estimates of their decoding failure rates, which, confirmed by experimental results, are almost the same as that of decoding Fqm-linear LRPC codes.

Due to their weak algebraic structure, low rank parity check (LRPC) codes have been employed in several post-quantum cryptographic schemes. In this paper we propose new improved decoding algorithms for ${\left[ {n,{\text{ }}k} \right]_{{q^m}}}$ LRPC codes of dual rank weight d. The proposed algorithms can efficiently decode LRPC codes with the parameters satisfying n − k = rd − c, where r is the dimension of the error support and c ≤ d − 2. They outperform the original decoding algorithm of LRPC codes when d > 2 and allow for decoding LRPC codes with a higher code rate and smaller values m.

In this work we propose a family of ${\mathbb{F}_q}$-linear lowrank parity check (LRPC) codes based on a bilinear product over $\mathbb{F}_q^m$ defined by a generic 3-tensor over ${\mathbb{F}_q}$. A particular choice of this tensor corresponds to the classical ${\mathbb{F}_{{q^m}}}$-linear LRPC codes; and other tensors yield ${\mathbb{F}_q}$-linear codes, which, with some caveats, can be efficiently decoded with the same idea of decoding LRPC codes. The proposed codes contribute to the diversity of rank metric codes for cryptographic applications, particularly for the cases where attacks utilize ${\mathbb{F}_{{q^m}}}$-linearity to reduce decoding complexity.