Linear codes with few weights have attracted significant interest due to their wide-ranging applications in secret sharing, authentication codes, association schemes, strongly regular graph, and some other fields. This paper focuses on unifying several existing construction methods for few-weight linear codes, extending the works of Wang et al. (2015) [24], Wu et al. (2019) [25], and Fang et al. (2023) [10]. In our code construction, we introduce a novel index set J, whose cardinality and structural properties are shown to critically influence both the length and weight distribution of the resulting few-weight linear codes. By employing cyclotomic mappings and choosing the more general defining sets, several new classes of binary linear codes with at most three weights are constructed. Our framework subsumes all aforementioned constructions as special cases and enlarges the spectrum of attainable parameters. The weight distributions of the corresponding linear codes are also explicitly determined. We also demonstrate that some of the linear codes constructed in this paper are optimal in the sense that they have the best known parameters in the tables maintained by Markus Grassl and/or optimal in the sense that they meet certain bounds on linear codes. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Delay invariant convolutional network codes (abbreviated DI-CNCs) can guarantee multicast communication at asymptotically optimal rates in networks with any delay profile. For a cyclic network, it has been shown that one can associate it with an acyclic network consisting of nodes in five layers, and the acyclic algorithm of $\mathbb {F}$ -linear multicast can be employed to construct a DI- $\mathbb {F}$ -CNC, as long as the field size is larger than the number of sinks in the cyclic network. In this paper, we present a directly feasible construction algorithm for a DI- $\mathbb {F}$ -CNC over a cyclic network. Complexity of code construction and theoretical guarantees of algorithm implementation are also investigated in detail. The advantage of the straight construction algorithm is that for an existing code, when some sink nodes and associated edges are added, our algorithm just modifies the new assigned coding coefficients in an efficient localized manner, without the necessity to construct again the code in its expanding network.

Binary sequences with high linear complexity and high 2-adic complexity have important applications in communication and cryptography. In this paper, the 2-adic complexity of a class of balanced Whiteman generalized cyclotomic sequences which have high linear complexity is considered. Through calculating the determinant of the circulant matrix constructed by one of these sequences, the result shows that the 2-adic complexity of this class of sequences is large enough to resist the attack of the rational approximation algorithm (RAA) for feedback with carry shift registers (FCSRs).

基于"以学生为中心"的教育理念,本文从教学资源、教学内容、教学方法与模式及考核评价四方面系统总结了我校对高等数学课程实施的教学改革.改革和实践结果表明,丰富且与时俱进的线上、线下教学资源可满足学生个性化学习需求,激励学生自主学习;凸显课程内涵、外拓实际应用,结合学生专业的优质教学内容可提高学生的学习积极性和应用创新能力;以趣为导、情感为基、合作探究的教学方法可激发学生兴趣,提高学生学习参与度;多元化、过程化的考评方式可提升学生的综合素养,促使学生养成良好的学习习惯.

In this paper, we study Hermitian linear complementary dual (abbreviated Hermitian LCD) rank metric codes. A class of Hermitian LCD generalized Gabidulin codes are constructed by q(m)-self-dual bases of F-q2m over F-q2. Moreover, the exact number of q(m)-self-dual bases of F-q2m over F-q2 is derived. As a consequence, an upper bound and a lower bound of the number of the constructed Hermitian LCD generalized Gabidulin codes are determined.

本文利用欧拉公式给出两类特殊类型函数的不定积分解法.

Linear complementary dual (abbreviated LCD) generalized Gabidulin codes (including Gabidulin codes) have been recently investigated by Shi and Liu et al. (Shi et al. IEICE Trans. Fundamentals E101-A(9):1599-1602, 2018, Liu et al. Journal of Applied Mathematics and Computing 61(1): 281-295, 2019). They have constructed LCD generalized Gabidulin codes of length $ n $ over $ \mathbb{F}_{q^{n}} $ by using self-dual bases of $ \mathbb{F}_{q^{n}} $ over $ \mathbb{F}_{q} $ when $ q $ is even or both $ q $ and $ n $ are odd. Whereas for the case of odd $ q $ and even $ n $, whether LCD generalized Gabidulin codes of length $ n $ over $ \mathbb{F}_{q^{n}} $ exist or not is still open. In this paper, it is shown that one can always construct LCD generalized Gabidulin codes of length $ n $ over $ \mathbb{F}_{q^{n}} $ for the case of odd $ q $ and even $ n $.

从函数改变量的微分近似出发,通过类比归纳和误差"捡回"的方法导入泰勒公式,并阐述其几何意义、物理意义、原型、唯一性、缺点,最后从哲学的角度分析泰勒公式蕴含的真善美.既有助于学生理解和掌握泰勒公式,又能提升学生的数学思维品质.

In this paper, we study a family of constacyclic BCH codes over $\mathbb{F}_{q^2}$ of length $n=\frac{q^{2m}-1}{q+1}$, where $q$ is a prime power, and $m\geq2$ an even integer. The maximum design distance of narrow-sense Hermitian dual-containing constacyclic BCH codes of length $n$ is determined. Furthermore, the exact dimension of the constacyclic BCH codes with given design distance is computed. As a consequence, we are able to derive the parameters of quantum codes as a function of their design parameters of the associated constacyclic BCH codes. This improves the result by Yuan et al. (Des Codes Cryptogr 85(1): 179-190, 2017), showing that with the same lengths, except for three trivial cases ($q=2,3,4$), our resultant quantum codes can always yield strict dimension or minimum distance gains than the ones obtained by Yuan et al.. Moreover, fixing length $n=\frac{q^{2m}-1}{q+1}$, some constructed quantum codes have better parameters or are beneficial complements compared with some known results (Aly et al., IEEE Trans Inf Theory 53(3): 1183-1188, 2007, Li et al., Quantum Inf Process 18(5): 127, 2019, Wang et al., Quantum Inf Process 18(8): 323, 2019, Song et al., Quantum Inf Process 17(10): 1-24, 2018.).

本文首先利用矩阵与其特征值的关系分析平面内二次曲线的形状;进而利用正交变换的方法将圆柱面与平面斜交产生椭圆的经验结论从数学本身的严谨性出发给出确切的公式说明,并给出椭圆的两个半轴长,焦点及顶点坐标的具体表达式;最后针对椭圆柱面与平面斜交可以产生圆的问题进行剖析,从而找出能使平面与椭圆柱面相交成圆的具体的倾斜方向.

Using inverse Gray mapping and generalized cyclotomic binary sequences, Zheng Yang et al.[2] constructed a quaternary sequence of length pq with low autocorrelation. In this paper, we present a new quaternary sequence by a variant of inverse Gray mapping and the generalized cyclotomic binary sequences, and calculate autocorrelation function of the new quaternary sequence. The results show that these two quaternary sequences have the same maximum nontrivial autocorrelation magnitude. Therefore it provides more quaternary sequences with low autocorrelation in communication systems.

In this letter, we introduce a new technique for deriving the exact decoding delay distribution expressions of random linear coding schemes over a lossy channel. Our approach is mainly based on two transformations: on the one hand, from an overall perspective, the correspondence between the calculation of the exact decoding delay distribution over a perfect channel and the counting problem for some special matrices can be established. On the other hand, the relationship between some cardinalities of the special matrices and the typical basis of polynomials can be obtained. Last, the decoding delay distribution issues can be solved by employing the polynomial basis representation method.

以同济大学应用数学系编的《高等数学》第六版一道例题为例,给出其另一种解法,并以此方法解决一类定积分的计算,既丰富了学生的知识,又开拓了学生的思维,这有利于高等数学教材的建设.

In this letter, we present an improved method for the independence test procedure in the convolutional multicast algorithm proposed by Erez and Feder. We employ the linear independence test vectors to check the independence of the partial encoding vectors in the main program of Erez's convolutional multicast algorithm. It turns out that compared with the previous approach of computing the determinants of the correlative matrices, carrying out the independence test vectors can reduce the computational complexity.

通过对几道不等式证明题的分析,文中总结了利用微分学证明不等式的常用方法,有助于学生加深对微分学知识的理解,激发学生的学习兴趣,也有利于学生发散思维培养及提高解决问题的能力.

通过对几道不定积分例题细致分析,阐明凑法可简化特殊类型函数的不定积分的解题过程,达到事半功倍的效果,也有利于提高学生灵活解题的能力和激发他们的学习兴趣.

A facile strategy was established to develop a drug delivery system (DDS) based on the graphene oxide nanoparticles (GON) with suitable size and shape to deliver drug effectively, by grafting the biocompatible PEGylated alginate (ALG-PEG) brushes onto the GON via the disulfide bridge bond. TEM analysis and drug-loading performance revealed that the 3-D nanoscaled, biocompatible, reduction-responsive nanocarriers (GON-Cy-ALG-PEG) were spherical in shape with diameters of 94.73 ± 9.56 nm. They possessed high doxorubicin (DOX)-loading capacity and excellent encapsulation efficiency, owing to their unique 3-D nanoscaled structure. They also had excellent stability in simulated physiological conditions and remarkable biocompatibility. Importantly, the in vitro release showed that the platform could not only prevent the leakage of the loaded DOX under physiological conditions but also detach the cytamine (Cy) modified PEGylated alginate (Cy-ALG-PEG) moieties, response to glutathione (GSH). Confocal microscopy and WST-1 assays provided clear evidence of the DOX-loaded GON-Cy-ALG-PEG endocytosis, whereas the drug-loaded nanocarriers exhibited high cytotoxicity to model cells. Furthermore, the cell apoptosis also was monitored via Flow cytometry. The results indicated that the DOX-loaded nanocarriers presented favorable efficiency of cell apoptosis. So these findings demonstrate that the accelerated release of the loaded DOX was realized in the presence of an elevated GSH that simulate the acidic endosomal compartments.

Under random linear coding (RLC) scheme, we present a simple expression of the probability distribution p(D = K) for decoding delay D incurred by the lossy channel, where K is a positive integer. In contrast with the previous contribution, our focus is firstly on deriving the cumulative distribution function of the discrete random variable D over a perfect channel. One benefit of such dispose is that, from the overall viewpoint, computing the cumulative distribution function of delay D can be related with calculating the cardinalities of sets of some special matrices, so that the former can be obtained from the latter. Moreover, our expression of the probability distribution is an explicit form, and is valid for any number of packets M, freewill field size q and arbitrary channel loss rate epsilon.

Similar to acyclic networks, over cyclic networks, there also exist four classes of optimal convolutional network codes, which are referred to as basic convolutional network code (BCNC), convolutional dispersion (CD), convolutional broadcast (CB), and convolutional multicast (CM), respectively. And from the perspective of linear independence among the global encoding kernels (GEKs), BCNC is with the best strength. In this paper, we present an efficient construction algorithm for BCNC over cyclic networks. Our algorithm can positively provide the maximal required cardinality of the local encoding kernels (LEKs). Another advantage of this algorithm is that for an existing code, when some non-source nodes and associated edges are added, our algorithm can correspondingly modify the already assigned LEKs in a localized manner. And we can just reset the LEKs along some special flow paths educed by the added nodes and edges, rather than reconstructing the whole code in its expanding network.