Abstract
Low-rank parity-check (LRPC) codes are rank-metric codes over finite fields, which have been proposed by Gaborit et al. (Proceedings of the workshop on coding and cryptography WCC, vol 2013, 2013) for cryptographic applications. Inspired by a recent adaption of Gabidulin codes to certain finite rings by Kamche et al. (IEEE Trans Inf Theory 65(12):7718–7735, 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.
Highlights
Rank-metric codes are sets of matrices whose distance is measured by the rank of their difference
The results of this paper were partly presented at the IEEE International Symposium on Information Theory 2020 [21]. Compared to this conference version, we generalize the results in two ways: first, we consider low-rank parity-check (LRPC) codes over the more general class of Galois rings instead of the integers modulo a prime power. This is a natural generalization since Galois rings share with finite fields many of the properties needed for dealing with the rank metric
We presented a failure probability bound for the decoder, whose derivation is significantly more involved than the finite-field analog due to the weaker structure of modules over finite rings
Summary
Rank-metric codes are sets of matrices whose distance is measured by the rank of their difference. The result uses a similar approach as [23] to transformation the channel output into a rank-metric error-erasure decoding problem Another application of rank-metric codes over finite rings are space-time codes. The results of this paper were partly presented at the IEEE International Symposium on Information Theory 2020 [21] Compared to this conference version, we generalize the results in two ways: first, we consider LRPC codes over the more general class of Galois rings instead of the integers modulo a prime power. This is a natural generalization since Galois rings share with finite fields many of the properties needed for dealing with the rank metric. We provide a complete analysis of the failure probability for all types of errors
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