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Institute
The Montgomery multiplication is an efficient method for modular arithmetic. Typically, it is used for modular arithmetic over integer rings to prevent the expensive inversion for the modulo reduction. In this work, we consider modular arithmetic over rings of Gaussian integers. Gaussian integers are subset of the complex numbers such that the real and imaginary parts are integers. In many cases Gaussian integer rings are isomorphic to ordinary integer rings. We demonstrate that the concept of the Montgomery multiplication can be extended to Gaussian integers. Due to independent calculation of the real and imaginary parts, the computation complexity of the multiplication is reduced compared with ordinary integer modular arithmetic. This concept is suitable for coding applications as well as for asymmetric key cryptographic systems, such as elliptic curve cryptography or the Rivest-Shamir-Adleman system.
Embodiments are generally related to the field of channel and source coding of data to be sent over a channel, such as a communication link or a data memory. Some specific embodiments are related to a method of encoding data for transmission over a channel, a corresponding decoding method, a coding device for performing one or both of these methods and a computer program comprising instructions to cause said coding device to perform one or both of said methods.
Method and device for error correction coding based on high-rate generalized concatenated codes
(2017)
Field error correction coding is particularly suitable for applications in non-volatile flash memories. We describe a method for error correction encoding of data to be stored in a memory device, a corresponding method for decoding a codeword matrix resulting from the encoding method, a coding device, and a computer program for performing the methods on the coding device, using a new construction for high-rate generalized concatenated (GC) codes. The codes, which are well suited for error correction in flash memories for high reliability data storage, are constructed from inner nested binary Bose-Chaudhuri-Hocquenghem (BCH) codes and outer codes, preferably Reed-Solomon (RS) codes. For the inner codes extended BCH codes are used, where only single parity-check codes are applied in the first level of the GC code. This enables high-rate codes.
A soft input decoding method and a decoder for generalized concatenated (GC) codes. The GC codes are constructed from inner nested block codes, such as binary Bose-Chaudhuri-Hocquenghem, BCH, codes and outer codes, such as Reed-Solomon, RS, codes. In order to enable soft input decoding for the inner block codes, a sequential stack decoding algorithm is used. Ordinary stack decoding of binary block codes requires the complete trellis of the code. In one aspect, the present invention applies instead a representation of the block codes based on the trellises of supercodes in order to reduce the memory requirements for the representation of the inner codes. This enables an efficient hardware implementation. In another aspect, there is provided a soft input decoding method and device employing a sequential stack decoding algorithm in combination with list-of-two decoding which is particularly well suited for applications that require very low residual error rates.
This work proposes a construction for low-density parity-check (LDPC) codes over finite Gaussian integer fields. Furthermore, a new channel model for codes over Gaussian integers is introduced and its channel capacity is derived. This channel can be considered as a first order approximation of the additive white Gaussian noise channel with hard decision detection where only errors to nearest neighbors in the signal constellation are considered. For this channel, the proposed LDPC codes can be decoded with a simple non-probabilistic iterative decoding algorithm similar to Gallager's decoding algorithm A.
The computational complexity of the optimal maximum likelihood (ML) detector for spatial modulation increases rapidly as more transmit antennas or larger modulation orders are employed. Hence, ML detection may be infeasible for higher bit rates. This work proposes an improved suboptimal detection algorithm based on the Gaussian approximation method. It is demonstrated that the new method is closely related to the previously published signal vector based detection and the modified maximum ratio combiner, but can improve the detection performance compared to these methods. Furthermore, the performance of different signal constellations with suboptimal detection is investigated. Simulation results indicate that the performance loss compared to ML detection depends heavily on the signal constellation, where the recently proposed Eisenstein integer constellations are beneficial compared to classical QAM or PSK constellations.
Multi-dimensional spatial modulation is a multipleinput/ multiple-output wireless transmission technique, that uses only a few active antennas simultaneously. The computational complexity of the optimal maximum-likelihood (ML) detector at the receiver increases rapidly as more transmit antennas or larger modulation orders are employed. ML detection may be infeasible for higher bit rates. Many suboptimal detection algorithms for spatial modulation use two-stage detection schemes where the set of active antennas is detected in the first stage and the transmitted symbols in the second stage. Typically, these detection schemes use the ML strategy for the symbol detection. In this work, we consider a suboptimal detection algorithm for the second detection stage. This approach combines equalization and list decoding. We propose an algorithm for multi-dimensional signal constellations with a reduced search space in the second detection stage through set partitioning. In particular, we derive a set partitioning from the properties of Hurwitz integers. Simulation results demonstrate that the new algorithm achieves near-ML performance. It significantly reduces the complexity when compared with conventional two-stage detection schemes. Multi-dimensional constellations in combination with suboptimal detection can even outperform conventional signal constellations in combination with ML detection.
This work proposes a suboptimal detection algorithm for generalized multistream spatial modulation. Many suboptimal detection algorithms for spatial modulation use two-stage detection schemes where the set of active antennas is detected in the first stage and the transmitted symbols in the second stage. For multistream spatial modulation with large signal constellations the second detection step typically dominates the detection complexity. With the proposed detection scheme, the modified Gaussian approximation method is used for detecting the antenna pattern. In order to reduce the complexity for detecting the signal points, we propose a combined equalization and list decoding approach. Simulation results demonstrate that the new algorithm achieves near-maximum-likelihood performance with small list sizes. It significantly reduces the complexity when compared with conventional two-stage detection schemes.
List decoding for concatenated codes based on the Plotkin construction with BCH component codes
(2021)
Reed-Muller codes are a popular code family based on the Plotkin construction. Recently, these codes have regained some interest due to their close relation to polar codes and their low-complexity decoding. We consider a similar code family, i.e., the Plotkin concatenation with binary BCH component codes. This construction is more flexible regarding the attainable code parameters. In this work, we consider a list-based decoding algorithm for the Plotkin concatenation with BCH component codes. The proposed list decoding leads to a significant coding gain with only a small increase in computational complexity. Simulation results demonstrate that the Plotkin concatenation with the proposed decoding achieves near maximum likelihood decoding performance. This coding scheme can outperform polar codes for moderate code lengths.