Weight distribution of two classes of cyclic codes with respect to two distinct order elements

Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. Cyclic codes have been studied for many years, but their weight distribution are known only for a few cases. In this paper, let $\Bbb F_r$ be an extension of a finite field $\Bbb F_q$ and $r=q^m$, we determine the weight distribution of the cyclic codes $\mathcal C=\{c(a, b): a, b \in \Bbb F_r\},$ c(a, b)=(\mbox {Tr}_{r/q}(ag_1^0+bg_2^0), \ldots, \mbox {Tr}_{r/q}(ag_1^{n-1}+bg_2^{n-1})), g_1, g_2\in \Bbb F_r, in the following two cases: (1) \ord(g_1)=n, n|r-1 and $g_2=1$; (2) \ord(g_1)=n, $g_2=g_1^2$, \ord(g_2)=\frac n 2, $m=2$ and $\frac{2(r-1)}n|(q+1)$

Weight distributions of cyclic codes with respect to pairwise coprime order elements

Let $\Bbb F_r$ be an extension of a finite field $\Bbb F_q$ with $r=q^m$. Let each $g_i$ be of order $n_i$ in $\Bbb F_r^*$ and $\gcd(n_i, n_j)=1$ for $1\leq i \neq j \leq u$. We define a cyclic code over $\Bbb F_q$ by $$\mathcal C_{(q, m, n_1,n_2, ..., n_u)}=\{c(a_1, a_2, ..., a_u) : a_1, a_2, ..., a_u \in \Bbb F_r\},$$ where $$c(a_1, a_2, ..., a_u)=({Tr}_{r/q}(\sum_{i=1}^ua_ig_i^0), ..., {Tr}_{r/q}(\sum_{i=1}^ua_ig_i^{n-1}))$$ and $n=n_1n_2... n_u$. In this paper, we present a method to compute the weights of $\mathcal C_{(q, m, n_1,n_2, ..., n_u)}$. Further, we determine the weight distributions of the cyclic codes $\mathcal C_{(q, m, n_1,n_2)}$ and $\mathcal C_{(q, m, n_1,n_2,1)}$.Comment: 18 pages. arXiv admin note: substantial text overlap with arXiv:1306.527

Combining Low-dimensional Wavelet Features and Support Vector Machine for Arrhythmia Beat Classification

Automatic feature extraction and classification are two main tasks in abnormal ECG beat recognition. Feature extraction is an important prerequisite prior to classification since it provides the classifier with input features, and the performance of classifier depends significantly on the quality of these features. This study develops an effective method to extract low-dimensional ECG beat feature vectors. It employs wavelet multi-resolution analysis to extract time-frequency domain features and then applies principle component analysis to reduce the dimension of the feature vector. In classification, 12-element feature vectors characterizing six types of beats are used as inputs for one-versus-one support vector machine, which is conducted in form of 10-fold cross validation with beat-based and record-based training schemes. Tested upon a total of 107049 beats from MIT-BIH arrhythmia database, our method has achieved average sensitivity, specificity and accuracy of 99.09%, 99.82% and 99.70%, respectively, using the beat-based training scheme, and 44.40%, 88.88% and 81.47%, respectively, using the record-based training scheme

(μ-4,4′-Bipyridine-κ2 N:N′)bis­[bis­(N,N-dimethyl­dithio­carbamato-κ2 S,S′)zinc(II)]

The title dinuclear ZnII complex, [Zn2(C3H6NS2)4(C10H8N2)], is centrosymmetric; the mid-point of the C—C bond linking the two pyridine rings is located on an inversion center. The pyridine N atom coordinates to the ZnII cation, which is also chelated by two dimethyl­dithio­carbamate anions, giving a trigonal-bipyramidal ZnNS4 geometry. Weak inter­molecular C—H⋯S hydrogen bonding is present in the crystal structure

Guaranteed Lower Eigenvalue Bound of Steklov Operator with Conforming Finite Element Methods

For the eigenvalue problem of the Steklov differential operator, by following Liu's approach, an algorithm utilizing the conforming finite element method (FEM) is proposed to provide guaranteed lower bounds for the eigenvalues. The proposed method requires the a priori error estimation for FEM solution to nonhomogeneous Neumann problems, which is solved by constructing the hypercircle for the corresponding FEM spaces and boundary conditions. Numerical examples are also shown to confirm the efficiency of our proposed method.Comment: 21 pages, 4 figures, 4 table