A Scaling Law to Predict the Finite-Length Performance of Spatially-Coupled LDPC Codes

Spatially-coupled LDPC codes are known to have excellent asymptotic properties. Much less is known regarding their finite-length performance. We propose a scaling law to predict the error probability of finite-length spatially-coupled ensembles when transmission takes place over the binary erasure channel. We discuss how the parameters of the scaling law are connected to fundamental quantities appearing in the asymptotic analysis of these ensembles and we verify that the predictions of the scaling law fit well to the data derived from simulations over a wide range of parameters. The ultimate goal of this line of research is to develop analytic tools for the design of spatially-coupled LDPC codes under practical constraints

How to Find Good Finite-Length Codes: From Art Towards Science

We explain how to optimize finite-length LDPC codes for transmission over the binary erasure channel. Our approach relies on an analytic approximation of the erasure probability. This is in turn based on a finite-length scaling result to model large scale erasures and a union bound involving minimal stopping sets to take into account small error events. We show that the performances of optimized ensembles as observed in simulations are well described by our approximation. Although we only address the case of transmission over the binary erasure channel, our method should be applicable to a more general setting.Comment: 13 pages, 13 eps figures, enhanced version of an invited paperat the 4th International Symposium on Turbo Codes and Related Topics, Munich, Germany, 200

Tree-Structure Expectation Propagation for LDPC Decoding over the BEC

We present the tree-structure expectation propagation (Tree-EP) algorithm to decode low-density parity-check (LDPC) codes over discrete memoryless channels (DMCs). EP generalizes belief propagation (BP) in two ways. First, it can be used with any exponential family distribution over the cliques in the graph. Second, it can impose additional constraints on the marginal distributions. We use this second property to impose pair-wise marginal constraints over pairs of variables connected to a check node of the LDPC code's Tanner graph. Thanks to these additional constraints, the Tree-EP marginal estimates for each variable in the graph are more accurate than those provided by BP. We also reformulate the Tree-EP algorithm for the binary erasure channel (BEC) as a peeling-type algorithm (TEP) and we show that the algorithm has the same computational complexity as BP and it decodes a higher fraction of errors. We describe the TEP decoding process by a set of differential equations that represents the expected residual graph evolution as a function of the code parameters. The solution of these equations is used to predict the TEP decoder performance in both the asymptotic regime and the finite-length regime over the BEC. While the asymptotic threshold of the TEP decoder is the same as the BP decoder for regular and optimized codes, we propose a scaling law (SL) for finite-length LDPC codes, which accurately approximates the TEP improved performance and facilitates its optimization

Spherical and Hyperbolic Toric Topology-Based Codes On Graph Embedding for Ising MRF Models: Classical and Quantum Topology Machine Learning

The paper introduces the application of information geometry to describe the ground states of Ising models by utilizing parity-check matrices of cyclic and quasi-cyclic codes on toric and spherical topologies. The approach establishes a connection between machine learning and error-correcting coding. This proposed approach has implications for the development of new embedding methods based on trapping sets. Statistical physics and number geometry applied for optimize error-correcting codes, leading to these embedding and sparse factorization methods. The paper establishes a direct connection between DNN architecture and error-correcting coding by demonstrating how state-of-the-art architectures (ChordMixer, Mega, Mega-chunk, CDIL, ...) from the long-range arena can be equivalent to of block and convolutional LDPC codes (Cage-graph, Repeat Accumulate). QC codes correspond to certain types of chemical elements, with the carbon element being represented by the mixed automorphism Shu-Lin-Fossorier QC-LDPC code. The connections between Belief Propagation and the Permanent, Bethe-Permanent, Nishimori Temperature, and Bethe-Hessian Matrix are elaborated upon in detail. The Quantum Approximate Optimization Algorithm (QAOA) used in the Sherrington-Kirkpatrick Ising model can be seen as analogous to the back-propagation loss function landscape in training DNNs. This similarity creates a comparable problem with TS pseudo-codeword, resembling the belief propagation method. Additionally, the layer depth in QAOA correlates to the number of decoding belief propagation iterations in the Wiberg decoding tree. Overall, this work has the potential to advance multiple fields, from Information Theory, DNN architecture design (sparse and structured prior graph topology), efficient hardware design for Quantum and Classical DPU/TPU (graph, quantize and shift register architect.) to Materials Science and beyond.Comment: 71 pages, 42 Figures, 1 Table, 1 Appendix. arXiv admin note: text overlap with arXiv:2109.08184 by other author

Limites práticos de segurança da distribuição de chaves quânticas de variáveis contínuas

Discrete Modulation Continuous Variable Quantum Key Distribution (DM-CV-QKD) systems are very attractive for modern quantum cryptography, since they manage to surpass all Gaussian modulation (GM) system’s disadvantages while maintaining the advantages of using CVs. Nonetheless, DM-CV-QKD is still underdeveloped, with a very limited study of large constellations. This work intends to increase the knowledge on DM-CV-QKD systems considering large constellations, namely M-symbol Amplitude Phase Shift Keying (M-APSK) irregular and regular constellations. As such, a complete DM-CV-QKD system was implemented, con sidering collective attacks and reverse reconciliation under the realistic scenario, assuming Bob detains the knowledge of his detector’s noise. Tight security bounds were obtained considering M-APSK constellations and GM, both for the mutual information between Bob and Alice and the Holevo bound between Bob and Eve. M-APSK constellations with binomial distribution can approximate GM’s results for the secret key rate. Without the consideration of the finite size effects (FSEs), the regular constellation 256-APSK (reg. 32) with binomial distribution achieves 242.9 km, only less 7.2 km than GM for a secret key rate of 10¯⁶ photons per symbol. Considering FSEs, 256-APSK (reg. 32) achieves 96.4% of GM’s maximum transmission distance (2.3 times more than 4-PSK), and 78.4% of GM’s maximum compatible excess noise (10.2 times more than 4-PSK). Additionally, larger constellations allow the use of higher values of modulation variance in a practical implementation, i.e., we are no longer subjected to the sub-one limit for the mean number of photons per symbol. The information reconciliation step considering a binary symmetric channel, the sum-product algorithm and multi-edge type low den sity parity check matrices, constructed from the progressive edge growth algorithm, allowed the correction of keys up to 18 km. The consideration of multidimensional reconciliation allows 256-APSK (reg. 32) to reconcile keys up to 55 km. Privacy amplification was carried out considering the application of fast Fourier transforms to the Toeplitz extractor, being unable of extracting keys for more than, approximately, 49 km, almost haft the theoretical value, and for excess noises larger than 0.16 SNU, like the theoretical value.Os sistemas de distribuição de chaves quânticas com variáveis contínuas e modulação discreta (DM-CV-QKD) são muito atrativos para a criptografia quântica moderna, pois conseguem superar todas as desvantagens do sistema com modulação Gaussiana (GM) enquanto mantêm as vantagens do uso de CVs. No entanto, DM-CV-QKD ainda está subdesenvolvida, sendo o estudo de grandes constelações muito reduzido. Este trabalho pretende aumentar o conhecimento sobre os sistemas DM-CV-QKD com constelações grandes, nomeadamente as do tipo M-symbol Amplitude Phase Shift Keying (M-APSK) irregulares e regulares. Com isto, foi simulado um sistema DM-CV-QKD completo, considerando ataques coletivos e reconciliação reversa tendo em conta o cenário realista, assumindo que o Bob co nhece o ruído de seu detetor. Os limites de segurança foram obtidos considerando constelações M-APSK e GM, tanto para a informação mútua entre o Bob e a Alice, quanto para o limite de Holevo entre o Bob e a Eve. As constelações M-APSK com distribuição binomial aproximam-se à GM quanto à taxa de chave secreta. Sem considerar o efeito de tamanho finito (FSE), a constelação regular 256-APSK (reg. 32) com distribuição binomial atinge 242.9 km, apenas menos 7.2 km do que GM para uma taxa de chave secreta de 10¯⁶ fotões por símbolo. Considerando FSEs, a 256-APSK (reg. 32) atinge 96.4% da distância máxima de transmissão para GM (2.3 vezes mais que a 4-PSK), e 78.4% do valor máximo de excesso de ruído compatível para GM (10.2 vezes mais do que a 4-PSK). Adicionalmente, grandes constelações permitem o uso de valores mais altos de variância de modulação em implementações práticas, pelo que deixa de ser necessário um número de fotões por símbolo abaixo de um. A etapa de reconciliação de informação considerou um canal binário simétrico, o algoritmo soma-produto e matrizes multi-edge type low density parity check, construídas a partir do algoritmo progressive edge growth, permitindo a correção de chaves até 18 km. A consideração de reconciliação multidimensional permite que a 256-APSK (reg. 32) reconcilie chaves até 55 km. A amplificação de privacidade foi realizada considerando a aplicação de transformadas de Fourier rápidas ao extrator de Toeplitz, mostrando-se incapaz de extrair chaves para mais de, aproximadamente, 49 km, quase metade do valor teórico, e para excesso de ruído superior a 0.16 SNU, semelhante ao valor teórico.Mestrado em Engenharia Físic