Generalized low rank parity check codes

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.