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

A Empirical Analysis on Performance Evaluation of the Tertiary Industry in Eastern Chinese Province Based on DEA

This paper comprehensively evaluated the performance of the tertiary industry in eastern Chinese province, using Data envelopment Analysis and the statistical section data of tertiary industry status investigation. The result figures out the provinces which are DEA efficient and which are not, in eastern Chinese province at 2009. According the result and discussion, imbalanced regional development exists in eastern. So, the adjustment direction and range are given based on the DEA evaluation, that the development of technology-intensive industry should be the key of tertiary industry

Correlation Analysis Between Scientific and Technological Progress and Economic Development

There exists an interdependence between scientific and technological progress and economic development. The development of science and technology brings along economic development, meanwhile economic development offers powerful support for it. Based on statistical analysis of the relationship between Sci-Tech investment and economic growth in recent decade, this article explores the dependencies between them, and gives us a measurement model. The result suggests that local Sci-Tech investment is more effectively for GDP growth. Furthermore, a win-win proposal of seeking for Sci-Tech progress and economic growth is given.Key words: Science and technology development; Independent innovation; National innovation system; Measurement mode

Study on the Performance Evaluation of Tertiary Industry in China

In this paper, we collected the statistical data describing the development of tertiary industry for each province of China. The statistical section data from above data set, Data Envelopment Analysis Model is used for the performance evaluation of China’s tertiary industry. The result figures out the provinces of China which are DEA efficient and which are not in tertiary industry by 2009. Meanwhile, the result gives  advice on the direction adjustment and range for each province, that the development of technology-intensive industry should be the key factor of tertiary industry

Recent progress on weight distributions of cyclic codes over finite fields

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. In coding theory it is often desirable to know the weight distribution of a cyclic code to estimate the error correcting capability and error probability. In this paper, we present the recent progress on the weight distributions of cyclic codes over finite fields, which had been determined by exponential sums. The cyclic codes with few weights which are very useful are discussed and their existence conditions are listed. Furthermore, we discuss the more general case of constacyclic codes and give some equivalences to characterize their weight distributions

Four infinite families of ternary cyclic codes with a square-root-like lower bound

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. Inspired by the recent work on binary cyclic codes published in IEEE Trans. Inf. Theory, vol. 68, no. 12, pp. 7842-7849, 2022, and the arXiv paper arXiv:2301.06446, the objectives of this paper are the construction and analyses of four infinite families of ternary cyclic codes with length $n=3^m-1$ for odd $m$ and dimension $k \in \{n/2, (n + 2)/2\}$ whose minimum distances have a square-root-like lower bound. Their duals have parameters $[n, k^\perp, d^\perp]$, where $k^\perp \in \{n/2, (n- 2)/2\}$ and $d^\perp$ also has a square-root-like lower bound. These families of codes and their duals contain distance-optimal cyclic codes

Performance evaluation of on-chip wavelength conversion based on InP/In1−x_{1-x}Gax_xAsy_yP1−y_{1-y} semiconductor waveguide platforms

We propose and design the high confinement InP/In1-xGaxAsyP1-y semiconductor waveguides and report the results of effective wavelength conversion based on this platform. Efficient confinement and mode field area fluctuation at different wavelength is analyzed to achieve the high nonlinear coefficient. The numerical results show that nearly zero phase-mismatch condition can be satisfied through dispersion tailoring of InP/In1-xGaxAsyP1-y waveguides, and the wavelength conversion ranging over 40 nm with the maximum conversion efficiency -26.3 dB is achieved for fixing pump power 100 mW. Meanwhile, the influences of the doping parameter y and pumping wavelength on the bandwidth and conversion efficiency are also discussed and optimized. It is indicated the excellent optical properties of the InP/In1-xGaxAsyP1-y waveguides and pave the way towards direct integration telecom band devices on stand semiconductor platforms.Comment: 21 page