Error Correction Code

Discovered by Bose, Chaudhuri and Hocquenghem , , the BCH family of error correcting codes are one of the most studied families in coding theory. They are also among the best performing codes, particularly when the number of errors being corrected is small relative to the code length. In this article we consider binary codes with minimum distance of 7. We construct new families of codes with these BCH parameters via a generalisation of the Kasami-Welch Theorem. Now letting w +w 2 -k = r, we obtain the following equation, with 16 solutions in r which form an additive group as w + w 2 -k is 2-to-1 and linear. where A, B, C and D are functions of b, c and v, that is, they are fixed elements of the field. Now, re-apply the identity (1) for the above equation with three solutions r, s and r + s (where s will be a fixed non zero solution) to obtain

Discovered by Bose, Chaudhuri and Hocquenghem , , the BCH family of error correcting codes are one of the most studied families in coding theory. They are also among the best performing codes, particularly when the number of errors being corrected is small relative to the code length. In this article we consider binary codes with minimum distance of 7. We construct new families of codes with these BCH parameters via a generalisation of the Kasami-Welch Theorem. Now letting w +w 2 -k = r, we obtain the following equation, with 16 solutions in r which form an additive group as w + w 2 -k is 2-to-1 and linear. where A, B, C and D are functions of b, c and v, that is, they are fixed elements of the field. Now, re-apply the identity (1) for the above equation with three solutions r, s and r + s (where s will be a fixed non zero solution) to obtain

Graph bundles generalize the notion of covering graphs and graph products. In Imrich et al. (Discrete Math. 167=168 (1998) 393) authors constructed an algorithm that ÿnds a presentation as a nontrivial cartesian graph bundle for all graphs that are cartesian graph bundles over triangle-free simple base using the relation * having the square property. An equivalence relation R on the edge set of a graph has the (unique) square property if and only if any pair of adjacent edges which belong to distinct R-equivalence classes span exactly one induced 4-cycle (with opposite edges in the same R-equivalence class). In this paper we deÿne the unique square property and show that any weakly 2-convex equivalence relation possessing the unique square property determines the fundamental factorization of a graph as a nontrivial cartesian graph bundle over an arbitrary base graph, whenever it separates degenerate and nondegenerate edges of the factorization.

In ], an ultra-wide bandwidth time-hopping spread-spectrum code division multiple-access system employing a binary PPM signaling has been introduced, and its performance was obtained based on a Gaussian distribution assumption for the multiple-access interference. In this paper, we begin first by proposing to use a practical low-rate error correcting code in the system without any further required bandwidth expansion. We then present a more precise performance analysis of the system for both coded and uncoded schemes. Our analysis shows that the Gaussian assumption is not accurate for predicting bit error rates at high data transmission rates for the uncoded scheme. Furthermore, it indicates that the proposed coded scheme outperforms the uncoded scheme significantly, or more importantly, at a given bit error rate, the coding scheme increases the number of users by a factor which is logarithmic in the number of pulses used in time-hopping spread-spectrum systems.

In diffusion based molecular communication, the intersymbol interference (ISI) is an important reason for system performance degradation, which is caused by the random movement, out-of-order arrival and indistinguishability of the moleclues. In this paper, a new metric called crossover distance is introduced to measure the distance between the received bit sequence and the probably tranmitted bit sequences. A new decoding scheme of conventional codes is proposed based on crossover distance, which can enhance the communication reliability significantly. The theoretic analysis indicates that the proposed decoding algorithm provides an approximately maximal likelihood estimation of the information bits. The numerical results show that compared with uncoded systems and some existing channel codes, the proposed convolutional codes offer good performance with same throughputs.

In diffusion based molecular communication, the intersymbol interference (ISI) is an important reason for system performance degradation, which is caused by the random movement, out-of-order arrival and indistinguishability of the moleclues. In this paper, a new metric called crossover distance is introduced to measure the distance between the received bit sequence and the probably tranmitted bit sequences. A new decoding scheme of conventional codes is proposed based on crossover distance, which can enhance the communication reliability significantly. The theoretic analysis indicates that the proposed decoding algorithm provides an approximately maximal likelihood estimation of the information bits. The numerical results show that compared with uncoded systems and some existing channel codes, the proposed convolutional codes offer good performance with same throughputs.

The ancient difficulty for establishing a common cryptographic secret key between two communicating parties Alice and Bob is nicely summarized by the Catch-22 dictum of S.J. Lomonaco [1999], to wit: "in order to communicate in secret one must first communicate in secret". In other words, to communicate in secret, Alice and Bob must already have a shared secret key. In this paper we analyse an algorithm for establishing such a common secret key by public discussion, under the modest and practical requirement that Alice and Bob are initially in possession of keys A and B, respectively, of a common length N which are not necessarily equal but are such that the mutual information I(A, B) is non-zero. This assumption is tantamount to assuming only that the corresponding statistical variables are correlated. The common secret key distilled by the algorithm will enjoy perfect secrecy in the sense of Shannon. The method thus provides a profound generalization of traditional symmetric key cryptography and applies also to quantum cryptography. Here, by purely elementary methods, we give a rigorous proof that the method proposed by Bennett, Bessette, Brassard, Salvail, and Smolin will in general converge to a non-empty common key under moderate assumptions on the choice of block lengths provided the initial bit strings are sufficiently long. Full details on the length requirements are presented. Furthermore, we consider the question of which The research of the first and second authors is supported by grants from NSERC.

This report documents a five-step iterative experimental study aimed at enabling deterministic, property-based recall of knowledge-base entries from natural-language queries - building a Geometric Language Machine/Model.

Baseline diagnostics established that the incumbent SHA256-derived Golay encoding completely destroyed numeric proximity (mean Hamming distance ≈10.13/24 for near-identical inputs), making property-based retrieval impossible with the original pipeline. I designed and validated a Gray-code quantised numeric vector engine (117→142 bits) that preserves small numeric changes as small Hamming distances. Unified 166-bit representations (24 Golay + 142 Gray-code) were evaluated across four molecular representation strategies, revealing that stoichiometric additive composition fails for emergent molecular properties (e.g., water’s anomalously high boiling point). A five-step chat simulation demonstrated 7/7 targeted entries in the top-10 recall list across exact, partial, fuzzy, and qualitative query types, validating the proposed extension to the auto_trigger.py pipeline (not integrated into the UBP system).

The report concludes with a comprehensive discussion of successes, failures, and a roadmap for future development. This study was completed by K-Dense ai (as clearly branded) using my Universal Binary Principal (UBP) system.

Several evolutionary algorithms (EAs) applied to a wide class of communication network design problems modelled under the generalized Steiner problem (GSP) are evaluated. In order to provide a fault-tolerant design, a solution to this problem consists of a preset number of independent paths linking each pair of potentially communicating terminal nodes. This usually requires considering intermediate non-terminal nodes (Steiner nodes), which are used to ensure path redundancy, while trying to minimize the overall cost. The GSP is an NP-hard problem for which few algorithms have been proposed. This article presents a comparative study of pure and hybrid EAs applied to the GSP, codified over MALLBA, a general purpose library for combinatorial optimization. The algorithms were tested on several GSPs, and asset efficient numerical results are reported for both serial and distributed models of the evaluated algorithms.

In this paper we present a new Turbo analog error correcting coding scheme for real valued signals that are corrupted by impulsive noise. This Turbo code improves Donoho's deterministic construction by using a probabilistic approach. More specifically, our construction corrects more errors than the matrices of Donoho by allowing a vanishingly small probability of error (with the increase in block size). The problem of decoding the long block code is decoupled into two sets of parallel Linear Programming problems. This leads to a significant reduction in decoding complexity as compared to one-step Linear Programming decoding.

We present a counterexample to the conjecture of Bihan, Franz, McCrory, and van Hamel concerning the maximality of toric varieties. There exists a six dimensional projective toric variety X with the sum of the Z 2 Betti numbers of X(R) strictly less than the sum of the Z 2 Betti numbers of X(C).

The Gilbert-Varshamov bound (non-constructively) establishes the existence of binary codes of distance $1/2 -\epsilon$ and rate $\Omega(\epsilon^2)$ (where an upper bound of $O(\epsilon^2\log(1/\epsilon))$ is known). Ta-Shma [STOC 2017] gave an explicit construction of $\epsilon$-balanced binary codes, where any two distinct codewords are at a distance between $1/2 -\epsilon/2$ and $1/2+\epsilon/2$, achieving a near optimal rate of $\Omega(\epsilon^{2+\beta})$, where $\beta \to 0$ as $\epsilon \to 0$. We develop unique and list decoding algorithms for (essentially) the family of codes constructed by Ta-Shma. We prove the following results for $\epsilon$-balanced codes with block length $N$ and rate $\Omega(\epsilon^{2+\beta})$ in this family: - For all $\epsilon, \beta > 0$ there are explicit codes which can be uniquely decoded up to an error of half the minimum distance in time $N^{O_{\epsilon, \beta}(1)}$. - For any fixed constant $\beta$ independent of $\epsilon$, there is an ...

Forward error correction (FEC) plays an important role in the field of telecommunication and information theory as it improves the capacity of a channel. It has been observed that Reed Solomon Error Corrector is a powerful method for error detection and correction. Error Correction codes (ECC) are a mean of including redundancy in a stream of information bits to allow the detection and correction of symbol errors during transmission. This research paper describes new design strategy targeting area optimization in reconfigurable device (FPGA). Area optimization is a major issue as smaller components allow for better system adaption, which is an expected feature of reconfigurable system for space applications. The approach is applied in the design stage of a component for the communications module of an on-board computer system. The chosen component is a Reed-Solomon encoder, which has been implemented using a Hardware Description Language (VHDL) targeting an FPGA platform. The resear...

Single parity check (SPC) product codes are simple yet powerful codes that are used to correct errors and/or recover erasures. The focus of this paper is to evaluate the performance of such codes under erasure scenarios and to develop a tight upper bound for the post-decoding erasure rate. Closed form exact expressions are derived for up to seven erasures. The derived expressions are verified using exhaustive search. A bound is used to account for eight erasures and above. The developed expressions improve the evaluation of the recoverability of SPC product codes without the need for simulation or exhaustive search.

In this paper we consider the problem of finding bounds on the size of ternary and quaternary constant-weight equidistant codes with 2 ≤ w < n ≤ 10. Optimal ternary and quaternary constant-weight equidistant codes have been constructed by combinatorial and computer methods. All open cases for the size of optimal ternary and quaternary constant-weight equidistant codes have been closed.

References 1. L. Bortolussi and A. Sgarro. Possibilistic channels for dna word design. In Proceedings of SMPS 2006, 2006. 2. D. Dubois and H. Prade, editors. Fundamentals of Fuzzy Sets. Kluwer, 2000. 3. TL Fine. Theories of Probability. Academic Press, 1973. 4. A. Sgarro. ...

IBLT) is a probabilistic concise data structure for set representation that supports a listing operation as the recovery of the elements in the represented set. Its applications can be found in network synchronization and traffic monitoring as well as in error-correction codes. IBLT can list its elements with probability affected by the size of the allocated memory and the size of the represented set, such that it can fail with small probability even for relatively small sets. While previous works only studied the failure probability of IBLT, this work initiates the worst case analysis of IBLT that guarantees successful listing for all sets of a certain size. The worst case study is important since the failure of IBLT imposes high overhead. We describe a novel approach that guarantees successful listing when the set satisfies a tunable upper bound on its size. To allow that, we develop multiple constructions that are based on various coding techniques such as stopping sets and the stopping redundancy of error-correcting codes, Steiner systems, and covering arrays as well as new methodologies we develop. We analyze the sizes of IBLTs with listing guarantees obtained by the various methods as well as their mapping memory consumption. Lastly, we study lower bounds on the achievable sizes of IBLT with listing guarantees and verify the results in the paper by simulations. CCS Concepts: • Mathematics of computing → Coding theory; • Theory of computation → Data structures and algorithms for data management; Bloom filters and hashing; Data structures design and analysis.

Com a crescente demanda dos sistemas de telecomunicações e automatização, o crescimento na utilização de aplicações como compartilhamento de voz, dados e multimídia necessitam, cada vez mais, de sistemas de comunicações com alta velocidade em banda larga para transmissão de informações. A expressão Power Line Communication (PLC), também conhecida como Power Line Telecomunication (PLT) ou Broadband Communications over Power Lines(BPL) é empregada para identificar tecnologias, equipamentos, aplicações e serviços que proporcionem comunicações sem a necessidade de cabeamento adicional ao da rede elétrica. Originalmente, as linhas de energia elétrica foram criadas para transmissão de energia na faixa de freqüência 50-60Hz .Por esta razão, sinais indesejados como o ruído térmico, ruído impulsivo, provocado pela inserção e remoção de cargas ao longo da linha de energia e o ruído de banda estreita prejudicam bastante o sistema PLC. Por isso, o canal PLC constitue-se em um meio extremamente hostil para transmissão de dados, introduzindo ao sinal Interferência Intersimbólica (ISI), distorções e atenuação. Com o intuito de reduzir a ISI e diminuir as taxas de erro de bit (BER) do sinal transmitido, o uso de equalizadores neurais define-se como uma alternativa para canais não-lineares, variantes no tempo e com comportamento aleatório, como o canal PLC. A Rede Neural Artificial (RNA) do equalizador aprende e adapta-se constantemente ao comportamento do canal PLC, ajustando dinâmicamente os coeficientes do equalizador para que o mesmo aproxime o sinal transmitido do recebido na detecção do bit. O sistema PLC simulado computacionalmente com resultados apresentados neste artigo possui configuração inhouse, modulação em banda básica BPSK e equalizador neural MLP com algoritmo de treinamento baseado no gradiente descente. O ambiente de simulação computacional é o Matlab versão 7.0, utilizando a neural network toolbox como suporte para o equalizador neural.

In this paper we construct \begin{document}$ \mathbb{F}_2 $\end{document} -linear codes over \begin{document}$ \mathbb{F}_{2}^{b} $\end{document} with length \begin{document}$ n $\end{document} and dimension \begin{document}$ n-r $\end{document} where \begin{document}$ n = rb $\end{document} . These codes have good properties, namely cyclicity, low density parity-check matrices and maximum distance separation in some cases. For the construction, we consider an odd prime \begin{document}$ p $\end{document} , let \begin{document}$ n = p-1 $\end{document} and utilize a partition of \begin{document}$ \mathbb{Z}_n $\end{document} . Then we apply a Zech logarithm to the elements of these sets and use the results to construct an index array which represents the parity-check matrix of the code. These codes are always cyclic and the density of the parity-check and the generator matrices decreases to \begin{document}$ 0 $\end{document} as \begin{document}$ n $\end{document} grows (for a fixed...

Smart wireless techniques are at the core of many today's telecommunication and networked embedded systems where performance are enhanced by intertwining radio frequency (RF) and digital aspects. Therefore their design requires to focus on both domains. Traditional approaches for their simulation rely either on different domain-specific tools or on analog-mixed-signal modeling languages. In the former case, the simulation of the whole platform in the same session is not possible while in the latter case, simulation performance is limited by the computationally most intensive domain (usually RF). We present an extension of the SystemC Network Simulation Library that allows to simulate antenna details and node position together with digital hardware and software. The validation on a real wearable system shows that the proposed simulation approach achieves a good trade-off between accuracy and speed thus allowing fast exploration of various configurations in the early phase of the ...

In this paper further results on double ±1 error correcting codes over rings are presented. In particular optimal linear codes correcting ±1 type of errors over rings Ζ7 and Z9 are constructed. A method allowing to construct (2N, 2N-6) double ±1error correcting codes over rings Zm from the given (N, N-4) double ±1 error correcting codes over rings Zm is also developed.

Delays appear always more frequently in applications, ranging, e.g., from population dynamics to automatic control, where the study of steady states is undoubtedly of major concern. As many other dynamical systems, those generated by nonlinear delay equations usually obey the celebrated principle of linearized stability. Therefore, hyperbolic equilibria inherit the stability properties of the corresponding linearizations, the study of which relies on associated characteristic equations. The transcendence of the latter, due to the presence of the delay, leads to infinitely-many roots in the complex plane. Simple algebraic manipulations show, first, that all such roots belong to the intersection of two curves. Second, only one of these curves is crucial for stability, and relevant sufficient and/or necessary criteria can be easily derived from its analysis. Other aspects can be investigated under this framework and a link to the theory of modulus semigroups and monotone semiflows is also discussed.

One of the possible hypothesis about time is to consider time as fuzzy concept, in a way that two instants of time could be overlapped. Historically, some Mathematicians and Philosophers had a similar idea like Brouwer and Husserl . Throughout this article we show the positive impact of this change on Theory of Computation and Complexity Theory to rebuild it in a more successful and fruitful approach. We call this novel Theory TC*.

In this paper, an encoder and decoder system is proposed using Bose-Chaudhuri-Hocquenghem (BCH) doubleerror-correcting and triple-error detecting (DEC-TED) with emerging memories of low power and high decoding efficiency. An adaptive error correction technique and an invalid transition inhibition technique is enforced to the decoder. This is to improve the decoding efficiency and reduce the power consumption and delay. The adaptive error correction gives high decoding efficiency and invalid transition technique reduce the power consumption issue in conventional BCH decoders. The DEC-TED BCH decoder combines these two techniques by using a specific Error Correcting Code Clock and Flip Flops. This technique provides an error correcting encoder and decoder solution for low power and high-performance application using emerging memories. The design simulated in Xilinx FPGA using ISE Design Suite 14.5.

In this paper, we introduced a new type of Encryption and Error Correction scheme, which is called "A Combined Encryption and Turbo Coding Scheme: AES-TURBO". Although in previous studies error correction and encryption are handled independently, we combined error correction and Encryption functionality into one single step. This combined System's performances are evaluated in AWGN (Additive White Gaussian Noise) channel type. The results are compared with the system employing ideal encryption and decryption.

In this work, we propose structured Root-Low-Density Parity-Check (LDPC) codes and design techniques for block-fading channels. In particular, Quasi-Cyclic Root-LDPC codes, Irregular repeat-accumulate Root-LDPC codes and Controlled Doping Root-LDPC codes based on Progressive Edge Growth (PEG) techniques for block-fading channels are proposed. The proposed Root-LDPC codes are both suitable for channels under $F = 2, 3$ and $4$ independent fading per codeword. The performance of the proposed codes is investigated in terms of Frame Error Rate (FER). The proposed Root-LDPC codes are capable of achieving the channel diversity and outperform standard LDPC codes. For block-fading channel with $F = 2$ our proposed PEG-based Root-LDPC codes outperform PEG-based LDPC codes by $7.5$dB at a FER close to $10^{-3}$.

In this work, we propose structured Root-Check Low-Density Parity-Check (LDPC) codes and design techniques for block-fading channels. In particular, Quasi-Cyclic Root-Check LDPC codes, Irregular repeat-accumulate Root-Check LDPC codes and Controlled Doping Root-Check LDPC codes based on Progressive Edge Growth (PEG) techniques for block-fading channels are proposed. The proposed Root-Check LDPC codes are both suitable for channels under F = 2, 3 and 4 independent fading per codeword. The performance of the proposed codes is investigated in terms of Frame Error Rate (FER). The proposed Root-Check LDPC codes are capable of achieving the channel diversity and outperform standard LDPC codes. For block-fading channel with F = 2 our proposed PEG-based Root-Check LDPC codes outperform PEG-based LDPC codes by 7.5dB.

— Fault-tolerance and data fusion have been considered as two fundamental functions in wireless sensor networks. In this paper, we propose a novel approach for distributed multiclass classi cation using a fault-tolerant fusion rule for wireless sensor networks. Binary decisions from local sensors, possibly in the presence of faults, are forwarded to the fusion center that determines the nal classi cation result. Classi cation fusion in our approach is implemented via error correcting codes to incorporate fault-tolerance capability. This new approach not only provides an improved fault-tolerance capability but also reduces computation time and memory requirements at the fusion center. Code matrix design is essential for the design of such systems. Two ef cient code matrix design algorithms are proposed in this paper. The relative merits of both algorithms are also studied. We also develop suf cient conditions for asymptotic detection of the correct hypothesis by the proposed approach...

The ongoing development of wireless sensor networks (WSNs) demands not only low-power sensors and less system cost but also good performance. Considering this background, investigating a new technology to satisfy both requirements is an important issue for current development of wireless sensor network systems. In this paper, we consider the situation that the sensor nodes in WSNs are deployed in a harsh environment such that both channel fading and unexpected sensor faults may occur. This work then proposes to combine channel estimation and sensorfault protection using error-correcting coding technique so as to free the costly devices of channel estimation and equalization from fusion centers. Simulations, performed to examine the performance of our proposed blind-detection scheme, show that it can compete with the conventional training-sequence-based fusion in performance when the training sequences are retained for information bearing. By this new approach, the complexity in fusion centers can be reduced and the transmitted power of sensors decreased without sacrificing the performance.

Fault-tolerance and data fusion have been considered as two fundamental functions in wireless sensor networks. In this paper, we propose a novel approach for distributed multiclass classification using a fault-tolerant fusion rule for wireless sensor networks. Binary decisions from local sensors, possibly in the presence of faults, are forwarded to the fusion center that determines the final classification result. Classification fusion in our approach is implemented via error correcting codes to incorporate fault-tolerance capability. This new approach not only provides an improved fault-tolerance capability but also reduces computation time and memory requirements at the fusion center. Code matrix design is essential for the design of such systems. Two efficient code matrix design algorithms are proposed in this paper. The relative merits of both algorithms are also studied. We also develop sufficient conditions for asymptotic detection of the correct hypothesis by the proposed approach. Performance evaluation of the proposed approach in the presence of faults is provided. These results show significant improvement in fault-tolerance capability as compared with conventional parallel fusion networks.