Low-rank parity-check (LRPC) are rank-metric codes over finite fields, which
have been proposed by Gaborit et al. (2013) for cryptographic applications.
Inspired by a recent adaption of Gabidulin codes to certain finite rings by
Kamche et al. (2019), we define and study LRPC codes over Galois rings - a wide
class of finite commutative rings. We give a decoding algorithm similar to
Gaborit et al.'s decoder, based on simple linear-algebraic operations. We
derive an upper bound on the failure probability of the decoder, which is
significantly more involved than in the case of finite fields. The bound
depends only on the rank of an error, i.e., is independent of its free rank.
Further, we analyze the complexity of the decoder. We obtain that there is a
class of LRPC codes over a Galois ring that can decode roughly the same number
of errors as a Gabidulin code with the same code parameters, but faster than
the currently best decoder for Gabidulin codes. However, the price that one
needs to pay is a small failure probability, which we can bound from above.Comment: 37 pages, 1 figure, extended version of arXiv:2001.0480
