Euclidean Weight Research Articles

Analysis of PAM Mapping for Convolutional-Coded Physical-Layer Network Coding

In this letter, we investigate iterative decoding for convolutional-coded physical-layer network coding operated with bit-interleaved coded modulation (BICM-PNC) over two-way relay channel (TWRC). The superimposed constellation at the relay from two users may not remain the same characteristic as the original modulation at the users. Thus, we first propose a design of mapping for PAM modulations at the users and enable the superimposed constellation of the maximum squared Euclidean weight (MSEW). Then, we analyze the PNC mappings by means of an extrinsic information transfer (EXIT) chart. The analysis and simulation results show that the proposed mapping can provide a good match to a given outer channel code. In particular, it achieves much better error performance than the conventional semi-Gray and binary-bit mappings at high SNR region for both 8-PAM and 16-PAM modulations.

Formative Evaluation of College Students’ Online English Learning Based on Learning Behavior Analysis

To measure the achievements and progress of college students in online learning, online learning platforms and teachers must pay attention to the formative evaluation of the learning process. The relevant data should be fully utilized to analyze the online English learning behavior of college students, such that online learning platforms and teachers can make formative evaluation of the students’ online learning. However, the existing studies on formative evaluation are mostly theoretical. To solve the problem, this paper explores the formative evaluation of college students’ online English learning based on learning behavior analysis. Firstly, the density-based spatial clustering of applications with noise (DBSCAN) was adopted to analyze the data samples of college students’ online English learning behavior, the evaluation indices were selected for the formative evaluation of the said behavior, and the index weighting method was explained in details. Next, the school precaution function of online English learning was realized through the graph structure data prediction of students’ online learning behavior. Based on the proposed graph neural network, the clustering Euclidean distance weight was introduced to measure the similarity between two nodes. In addition, the weight update process was illustrated for the distance weight-based attention mechanism. The proposed formative evaluation approach was proved effective through experiments.

Non-iterative denoising algorithm for mechanical vibration signal using spectral graph wavelet transform and detrended fluctuation analysis

A novel non-iterative denoising technique combining spectral graph wavelet transform and detrended fluctuation analysis is proposed to solve the filtering problem of non-linear and non-stationary mechanical vibration signals. The vibration signal is firstly converted to a graph signal defined on the path graph. Then, the graph signal is decomposed into scaling function coefficients and spectral graph wavelet coefficients by spectral graph wavelet transform. Finally, the threshold is adopted to shrink the spectral graph wavelet coefficients, and the denoised signal is obtained via the inverse transform. Besides, an efficient criterion based on detrended fluctuation analysis is designed to select the decomposition level of spectral graph wavelet transform. The denoising performance of the presented approach for mechanical vibration signal has been thoroughly evaluated through analog signals compared with five conventional methods. The developed technique is then applied to denoise the vibration signal collected in hob fault experiments, and the influence of different optional parameters on the denoising performance is analyzed by single factor experiments. Experimental results indicate that the proposed approach can remove noise well and retain the fine signatures of signal as much as possible, and Euclidean weight function and Minimax threshold can achieve desired denoising capability when combined with soft threshold function or hard threshold function.

Using spectral Geodesic and spatial Euclidean weights of neighbourhood pixels for hyperspectral Endmember Extraction preprocessing

Spectral Mixture Analysis is one of the fundamental subjects encountered when dealing with remotely sensed hyperspectral images. Its goal is to identify constituent elements of mixed-pixels called Endmembers (EMs) and their associated abundance maps. In this paper, a novel Geodesic and Euclidean distances-based preprocessing (GEPP) is addressed which is coupled with the classical spectral-based EM Extraction algorithms (EEs). It combines both spatial and spectral information utilizing two approaches with the purpose of searching for spectrally pure and spatially homogenous pixels that may be identified as the EM candidates in subsequent EEs. GEPP reduces EE processing time by introducing a new correlation coefficient similarity function (CCSF) on the spectrally pure and spatially homogenous pixels pick up with the help of spectral weighting computations, unsupervised Fuzzy C-means (FCM) clustering algorithm and a spatial neighbourhood system using Markov Random Field (MRF) so that processing a large amount of mixed and heterogeneous pixels developed by EEs is avoided. Moreover, CCSF exploits the spatial Euclidean and novel spectral Geodesic weights to compute the final mean vector which is able to improve recognition of spatially homogenous regions that are highly spectrally correlated such that it leads to better results of unmixing accuracy. According to experimental results on three synthetic and four real hyperspectral scenes, hyperspectral unmixing outcomes are relatively improved in terms of SAD and RMSE-based error metrics and higher computation speed can be realized by our proposal in comparison with the state-of-the-arttechniques.

The extension theorem for the Lee and Euclidean weights over [formula omitted

The Lee and Euclidean weights have the extension property over the local rings Z/pkZ, p prime. The non-vanishing of certain Fourier coefficients is established by expressing the coefficients in terms of generalized Bernoulli numbers and making use of knowledge of the locations of zeros of Dirichlet L-functions.

The extension theorem for Lee and Euclidean weight codes over integer residue rings

The extension theorem is proved for the Lee and Euclidean weights over the integer residue ring $$\mathbb {Z}_m$$ , for $$m \ge 2$$ .

The extension problem for Lee and Euclidean weights

The extension problem is solved for the Lee and Euclidean weights over three families of rings of the form $\Z/N\Z$: $N=2^{\ell + 1}$, $N=3^{\ell + 1}$, or $N=p=2q+1$ with $p$ and $q$ prime. The extension problem is solved for the Euclidean PSK weight over $\Z/N\Z$ for all $N$.

MacWilliams type identities on the Lee and Euclidean weights for linear codes over [formula omitted

Motivated by the works of Shiromoto [3] and Shi et al. [4], we study the existence of the MacWilliams type identities with respect to the Lee and Euclidean weight enumerators for linear codes over the ring Zℓ. Necessary and sufficient conditions for the existence of the MacWilliams type identities with respect to the Lee and Euclidean weight enumerators for linear codes over Zℓ are given. Some examples about such MacWilliams type identities are also presented.

On the Covering Radius of Codes Over Z4 with Chinesh Euclidean Weight

On the Covering Radius of Codes Over Z4 with Chinesh Euclidean Weight

A note on a basic exact sequence for the Lee and Euclidean weights of linear codes over [formula omitted

This paper is devoted to presenting two counterexamples to Shiromoto's (1999) results [2] on MacWilliams type identities with respect to Lee weight enumerators and Euclidean weight enumerators over Zℓ.

Bit-Interleaved Coded Modulation Systems with Iterative Decoding and Partial Reusing QAM Signal Points

A new method of reusing a subset of an M-QAM signal constellation is presented for Bit-Interleaved Coded Modulation systems with Iterative Decoding (BICM-ID). In addition to the remapping of high-energy signals to low-energy signals in order to save the average signal energy as a shaping technique known so far, a new scheme of remapping low-energy signals to high-energy signals is proposed in order to gain the equivalent distance when decoding bits, often called the Squared Euclidean Weights (SEW). Numerical analysis and simulation results show that the partial reuse of the signals in the BICM-ID system with 16-QAM can improve the system performance in terms of lowering the bit error rate in the error floor region.

Optimal self-dual [formula omitted]-codes and a unimodular lattice in dimension 41

For lengths up to 47 except 37, we determine the largest minimum Euclidean weight among all Type I Z4-codes of that length. We also give the first example of an optimal odd unimodular lattice in dimension 41 explicitly, which is constructed from some Type I Z4-code of length 41.

Singleton's Bound in Euclidean Codes

There are three standard weight functions on a linear code viz. the Hamming weight, Lee weight and Euclidean weight. The Euclidean weight function is useful in connection with the lattice constructions, where the minimum norm of vectors in the lattice is related to the minimum Euclidean weight of the code. In this paper, we obtain Singleton's bound for Euclidean codes and introduce maximum Euclidean square distance separable (MESDS) codes.

An upper bound on the minimum weight of Type II [formula omitted]-codes

In this paper, we give a new upper bound on the minimum Euclidean weight of Type II Z 2 k -codes and the concept of extremality for the Euclidean weights when k = 3 , 4 , 5 , 6 . Together with the known result, we demonstrate that there is an extremal Type II Z 2 k -code of length 8 m ( m ⩽ 8 ) when k = 3 , 4 , 5 , 6 .

Reliability evaluation of conceptual colour design based on rough sets

Conceptual Colour Design (CCD) is a key step in product innovation. And to identify whether colours meet market demands is crucial for making right conceptual design. Thus a new reliability evaluation method of CCD is proposed in this paper. Firstly, a pure reliability evaluation model is presented according to the Euclidean distance between colour design ideas and market demand. Secondly, RS theory is introduced into calculating weights of influencing factors of colour market demand. The process of constructing decision table of colour RS and extracting weights is presented in detail. Thirdly, an integrated reliability model considering the Euclidean distance and weights of influencing factors is given. Finally, a CCD case of dress is analysed using this method. The results show the evaluation method is practical and effective.

AN UPPER BOUND ON THE NUMBER OF PARITY CHECKS FOR BURST ERROR DETECTION AND CORRECTION IN EUCLIDEAN CODES

There are three standard weight functions on a linear code viz. Hamming weight, Lee weight, and Euclidean weight. Euclidean weight function is useful in connection with the lattice constructions (2) where the minimum norm of vectors in the lattice is related to the min- imum Euclidean weight of the code. In this paper, we obtain an upper bound over the number of parity check digits for Euclidean weight codes detecting and correcting burst errors.

Sharp $p$-Divisibility of Weights in Abelian Codes Over ${\BBZ}/p^d{\BBZ}$

A theorem of McEliece on the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> -divisibility of Hamming weights in cyclic codes over F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> is generalized to Abelian codes over Zopf <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">/p</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sup> Zopf. This work improves upon results of Helleseth-Kumar-Moreno-Shanbhag, Calderbank-Li-Poonen, Wilson, and Katz. These previous attempts are not sharp in general, i.e., do not report the full extent of the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> -divisibility except in special cases, nor do they give accounts of the precise circumstances under which they do provide best possible results. This paper provides sharp results on <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> -divisibilities of Hamming weights and counts of any particular symbol for an arbitrary Abelian code over Zopf <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">/p</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sup> Zopf. It also presents sharp results on <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</i> -divisibilities of Lee and Euclidean weights for Abelian codes over Zopf <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">/4</i> Zopf.

Natural conjugate gradient training of multilayer perceptrons

Natural gradient (NG) descent, arguably the fastest on-line method for multilayer perceptron (MLP) training, exploits the “natural” Riemannian metric that the Fisher information matrix defines in the MLP weight space. It also accelerates ordinary gradient descent in a batch setting but then the Fisher matrix essentially coincides with the Gauss–Newton approximation of the Hessian of the MLP square error function and NG is thus related to the Levenberg–Marquardt (LM) method, which may explain its speed-up with respect to standard gradient descent. However, even this comparison is advantageous for NG descent as it should have a linear convergence in a Riemannian weight space compared to the superlinear one of the LM method in the Euclidean weight space. This suggests that it may be interesting to consider superlinear methods for MLP training in a Riemannian setting. In this work we shall discuss how to introduce a natural conjugate gradient (CG) method for MLP training. While a fully Riemannian formulation would result in an extremely costly procedure, we shall make some simplifying assumptions that should give descent directions with properties similar to those of standard CG descent. Moreover, we will also show numerically that natural CG may lead to a faster convergence to better minima, although with a greater cost than that of standard CG that, nevertheless, may be alleviated using a diagonal natural CG variant.

Soft-decision decoding of Reed-Muller codes: a simplified algorithm

Soft-decision decoding is considered for general Reed-Muller (RM) codes of length n and distance d used over a memoryless channel. A recursive decoding algorithm is designed and its decoding threshold is derived for long RM codes. The algorithm has complexity of order nlnn and corrects most error patterns of the Euclidean weight of order radicn/lnn, instead of the decoding threshold radicd/2 of the bounded distance decoding. Also, for long RM codes of fixed rate R, the new algorithm increases 4/pi times the decoding threshold of its hard-decision counterpart

Analysis and design of symbol mappers for iteratively decoded BICM

Iterative decoding and demodulation of bit-interleaved coded modulation (BICM) is investigated. The union bound on the error probability of BICM with maximum-likelihood (ML) decoding is derived with the assumption of a uniform interleaver. Based on the union bound, it is shown that an interleaving gain is achievable for BICM at high signal-to-noise ratio (SNR). The design rules on the symbol mapper for optimal asymptotic performance at high SNR are derived and a new symbol mapper, called the maximum squared Euclidean weight (MSEW) symbol mapper, is proposed with examples given for 8-PSK and 16-QAM. The MSEW mapper is shown by means of an extrinsic information transfer (EXIT) chart to provide the best error-floor performance for BICM at high SNR. At low SNR, a mapper doping technique is proposed that combines the MSEW mapper with Gray mapper to provide a good match to a given outer code. Simulation results are also presented to confirm our analysis.