An Algebraic Framework for End-to-End Physical-Layer Network Coding

Gorla, Elisa; Ravagnani, Alberto

doi:10.1109/tit.2017.2778726

Cited by 11 publications

(7 citation statements)

References 15 publications

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“…Now, multiplying both sides by g r −2 m and with the same reasoning as before, we obtain that all the λ ,i, j i ∈ m and the right-hand side of (11) belongs to m 3 . Iterating this process r − 2 times, we finally get that the right-hand side of (11) belongs to m r = (0), and therefore (11) corresponds to…”

Section: Lemma 6 Let M Be An R-submodule Of S and Let A B ∈ Free(m)mentioning

confidence: 99%

“…Recently, there has been an increased interest in rank-metric codes over finite rings due to the following applications. Network coding over certain finite rings was intensively studied in [ 7 , 11 ], motivated by works on nested-lattice-based network coding [ 8 , 18 , 26 , 28 ] which show that network coding over finite rings may result in more efficient physical-layer network coding schemes. Kamche et al [ 14 ] showed how lifted rank-metric codes over finite rings can be used for error correction in network coding.…”

Section: Introductionmentioning

confidence: 99%

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Low-rank parity-check codes over Galois rings

Renner

Neri

Puchinger

2020

Des. Codes Cryptogr.

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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.

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Section: Lemma 6 Let M Be An R-submodule Of S and Let A B ∈ Free(m)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Low-rank parity-check codes over Galois rings

Renner

Neri

Puchinger

2020

Des. Codes Cryptogr.

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“…m u ℓ a i,ji = 0, and since by hypothesis {u ℓ a i,ji } is a basis, this implies λ ℓ,i,ji ∈ Ann(g r−1 m ) = m and therefore there exist λ ′ ℓ,i,ji ∈ R, such that λ ℓ,i,ji = g m λ ′ ℓ,i,ji . Thus, (11) becomes…”

Section: ⊓ ⊔mentioning

confidence: 99%

“…Recently, there has been an increased interest in rank-metric codes over finite rings due to the following applications. Network coding over certain finite rings was intensively studied in [7,11], motivated by works on nested-lattice-based network coding [8,17,25,27] which show that network coding over finite rings may result in more efficient physical-layer network coding schemes. Kamche et al [13] showed how lifted rank-metric codes over finite rings can be used for error correction in network coding.…”

Section: Introductionmentioning

confidence: 99%

Low-Rank Parity-Check Codes over Galois Rings

Renner,

Neri,

Puchinger

2020

Preprint

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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.

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“…Network coding over finite rings [1]- [6] may result in more efficient physical-layer network coding schemes in comparison to using finite fields. Since rank-metric codes can be applied for error correction in network coding (cf.…”

Section: Introductionmentioning

confidence: 99%

Efficient Decoding of Gabidulin Codes over Galois Rings

Puchinger¹,

Renner²,

Wachter-Zeh³

et al. 2021

Preprint

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This paper presents the first decoding algorithm for Gabidulin codes over Galois rings with provable quadratic complexity. The new method consists of two steps: (1) solving a syndrome-based key equation to obtain the annihilator polynomial of the error and therefore the column space of the error, (2) solving a key equation based on the received word in order to reconstruct the error vector. This two-step approach became necessary since standard solutions as the Euclidean algorithm do not properly work over rings.

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An Algebraic Framework for End-to-End Physical-Layer Network Coding

Cited by 11 publications

References 15 publications

Low-rank parity-check codes over Galois rings

Low-rank parity-check codes over Galois rings

Low-Rank Parity-Check Codes over Galois Rings

Efficient Decoding of Gabidulin Codes over Galois Rings

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