Let H(D) be the parity-check matrix of an LDPC convolutional code corresponding to the parity-check matrix H of a QC code obtained using the method of Tanner et al. We see that the entries in H(D) are all monomials and several rows (columns) have monomial factors. Let us cyclically shift the rows of H. Then the parity-check matrix H'(D) corresponding to the modified matrix H' defines another convolutional code. However, its free distance is lower-bounded by the minimum distance of the original QC code. Also, each row (column) of H'(D) has a factor different from the one in H(D). We show that the state-space complexity of the error-trellis associated with H'(D) can be significantly reduced by controlling the row shifts applied to H with the error-correction capability being preserved.
We propose a decoder for Trellis-Constrained Codes, a super-class of Turbo- and LDPC codes. Inspired by amplitude amplification from quantum computing, we attempt to amplify the relative likelihood of the most likely codeword until it stands out from all other codewords.
On Existence of Girth-8 QC-LDPC Code with Large Column Weight: Combining Mirror-sequence with Classification Modulo Ten
Xi’an University of Posts and Telecommunications（西安邮电大学）；China Academy of Space Technology (Xi’an)（中国空间技术研究院（西安））；Guangdong University of Technology（广东工业大学）
This paper gives necessary and sufficient conditions for the Tanner graph of a quasi-cyclic (QC) low-density parity-check (LDPC) code based on the all-one protograph to have girth 6, 8, 10, and 12, respectively, in the case of parity-check matrices with column weight 4. These results are a natural extension of the girth results of the already-studied cases of column weight 2 and 3, and it is based on the connection between the girth of a Tanner graph given by a parity-check matrix and the properties of powers of the product between the matrix and its transpose. The girth conditions can be easi…
CC-GLPDC codes are a class of generalized low-density parity-check (GLDPC) codes where the constraint nodes (CNs) represent convolutional codes. This allows for efficient decoding in the trellis with the forward-backward algorithm, and the strength of the component codes easily can be controlled by the encoder memory without changing the graph structure. In this letter, we extend the class of CC-GLDPC codes by introducing different types of irregularity at the CNs and investigating their effect on the BP and MAP decoding thresholds for the binary erasure channel (BEC). For the considered class…
We give a construction of quantum LDPC codes of dimension $Θ(\log N)$ and distance $Θ(N/\log N)$ as the code length $N\to\infty$. Using a product of chain complexes this construction also provides a family of quantum LDPC codes of distance $Ω(N^{1-α/2}/\log N)$ and dimension $Ω(N^α\log N)$, where $0 \le α< 1$. We also introduce and study a new operation called lifted product, which naturally generalizes the product operations for quantum codes and chain complexes. Moreover, as a simple byproduct of our results on quantum codes, we obtain a new result on classical codes. We show that for any fi…
Low-Complexity SVD-Based Precoding over Zero-Padded ISI Channels
Cyber Security Research and Innovation Centre Holmes Institute（霍姆斯学院网络安全研究与创新中心）；Department of Software Systems and Cybersecurity FIT, Monash University（蒙纳士大学信息技术学院软件系统与网络安全系）
For an abelian group $H$ acting on the set $[\ell]$, an $(H,\ell)$-lift of a graph $G_0$ is a graph obtained by replacing each vertex by $\ell$ copies, and each edge by a matching corresponding to the action of an element of $H$. In this work, we show the following explicit constructions of expanders obtained via abelian lifts. For every (transitive) abelian group $H \leqslant \text{Sym}(\ell)$, constant degree $d \ge 3$ and $ε> 0$, we construct explicit $d$-regular expander graphs $G$ obtained from an $(H,\ell)$-lift of a (suitable) base $n$-vertex expander $G_0$ with the following parameters…
In this Phd thesis discusses modern methods for constructing MET QC-LDPC codes with a given error correction ("waterfall, error-floor") and complexity (parallelism level according circulant size plus scheduler orthogonality of checks) profiles: 1. weight enumerators optimization, protograph construction using Density Evolution, MI (P/Exit-chart) and it approximation: Gaussian Approximation, Reciprocal-channel approximation and etc; 2. Covariance evolution and it approximation; 3. Lifting methods for QC codes construction:PEG, Guest-and-Test, Hill-Climbing with girth, EMD, ACE optimization; 4. …
We introduce a new family of rank metric codes: Low Rank Parity Check codes (LRPC), for which we propose an efficient probabilistic decoding algorithm. This family of codes can be seen as the equivalent of classical LDPC codes for the rank metric. We then use these codes to design cryptosystems à la McEliece: more precisely we propose two schemes for key encapsulation mechanism (KEM) and public key encryption (PKE). Unlike rank metric codes used in previous encryption algorithms -notably Gabidulin codes - LRPC codes have a very weak algebraic structure. Our cryptosystems can be seen as an equi…
