Space-time codes based on rank-metric codes and their decoding

Puchinger, Sven; Stern, Sebastian; Bossert, Martin; Fischer, Robert F. H.

doi:10.1109/iswcs.2016.7600887

Cited by 11 publications

(13 citation statements)

References 18 publications

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“…When k > n/2 and λ > 1 the codes obtained by our constructions fall short of the upper bounds, unless k is close to n. An example for our weak bounds in this case can be demonstrated for n = 3ℓ, k = 2ℓ, t = ℓ + 1, and λ = 2. The upper bound for A q (3ℓ, 2ℓ, ℓ + 1; 2) by Proposition 1 is q ct 2 for some constant c. A probabilistic argument [71,75,78] yields that this bound is attained for smaller constant. But, there is no construction which is getting close to this value.…”

Section: Conclusion and Problems For Future Researchmentioning

confidence: 97%

Subspace packings: constructions and bounds

Etzion

Kurz

Otal

et al. 2020

Des. Codes Cryptogr.

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The Grassmannian G q (n, k) is the set of all k-dimensional subspaces of the vector space F n q . Kötter and Kschischang showed that codes in Grassmannian space can be used for error-correction in random network coding. On the other hand, these codes are qanalogs of codes in the Johnson scheme, i.e. constant dimension codes. These codes of the Grassmannian G q (n, k) also form a family of q-analogs of block designs and they are called subspace designs.In this paper, we examine one of the last families of q-analogs of block designs which was not considered before. This family called subspace packings is the q-analog of packings, and was considered recently for network coding solution for a family of multicast networks called the generalized combination networks. A subspace packing t-(n, k, λ) q is a set S of k-subspaces from G q (n, k) such that each t-subspace of G q (n, t) is contained in at most λ elements of S. The goal of this work is to consider the largest size of such subspace packings. We derive a sequence of lower and upper bounds on the maximum size of such packings, analyse these bounds, and identify the important problems for further research in this area.

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Section: Conclusion and Problems For Future Researchmentioning

confidence: 97%

Subspace packings: constructions and bounds

Etzion

Kurz

Otal

et al. 2020

Des. Codes Cryptogr.

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“…A variety of methods [3], [5]- [7], [23], [32], [35], [36] exist for obtaining space-time codes from codes over finite fields. We set aside those focused on binary fields because linearized Reed-Solomon codes are only interesting when nonbinary due to the q > L requirement.…”

Section: Background On Rank-metric-preserving Mapsmentioning

confidence: 99%

“…This leaves over the methods in [3]- [7]. In [6], the authors propose a framework which encompasses as special cases mappings to the Gaussian [3], [4] and Eisenstein [5] integers and find that, with the exception of when a small PSK constellation is required, the method in [7] is outperformed. We accordingly define a notion of rankmetric-preserving map only general enough to subsume these important special cases.…”

Section: Background On Rank-metric-preserving Mapsmentioning

confidence: 99%

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Space–Time Codes From Sum-Rank Codes

Shehadeh

Kschischang

2022

IEEE Trans. Inform. Theory

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Just as rank-metric or Gabidulin codes may be used to construct rate-diversity tradeoff optimal space-time codes, a recently introduced generalization for the sum-rank metriclinearized Reed-Solomon codes-accomplishes the same in the case of multiple fading blocks. In this paper, we provide the first explicit construction of minimal delay rate-diversity optimal multiblock space-time codes as an application of linearized Reed-Solomon codes. We also provide sequential decoders for these codes and, more generally, space-time codes constructed from finite field codes. Simulation results show that the proposed codes can outperform full diversity codes based on cyclic division algebras at low SNRs as well as utilize significantly smaller constellations.

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“…See the discussion in [28]. The diversity-rate tradeoff is expressed in a Singleton-type bound, and codes attaining equality in such a bound may be obtained by mapping a maximum rank distance (MRD) code over a finite field, such as a Gabidulin code, into the constellation A ⊆ C. This may be done via Gaussian integers [17] or Eisenstein integers [25].…”

Section: Space-time Coding With Multiple Fading Blocksmentioning

confidence: 99%

Maximum Sum-Rank Distance Codes over Finite Chain Rings

Martínez-Peñas¹,

Puchinger²

2021

Preprint

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In this work, maximum sum-rank distance (MSRD) codes and linearized Reed-Solomon codes are extended to finite chain rings. It is proven that linearized Reed-Solomon codes are MSRD over finite chain rings, extending the known result for finite fields. For the proof, several results on the roots of skew polynomials are extended to finite chain rings. These include the existence and uniqueness of minimumdegree annihilator skew polynomials and Lagrange interpolator skew polynomials. An efficient Welch-Berlekamp decoder with respect to the sum-rank metric is then provided for finite chain rings. Finally, applications in Space-Time Coding with multiple fading blocks and physical-layer multishot Network Coding are discussed.

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Space-time codes based on rank-metric codes and their decoding

Cited by 11 publications

References 18 publications

Subspace packings: constructions and bounds

Subspace packings: constructions and bounds

Space–Time Codes From Sum-Rank Codes

Maximum Sum-Rank Distance Codes over Finite Chain Rings

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