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ON THE COMPLEXITY OF THE NORMAL BASES VIA PRIME GAUSS PERIOD OVER FINITE FIELDS

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Journal of Systems Science & Complexity 2009年第3期22卷 395-406页

作者： Qunying LIAO keqin feng College of Mathematics and Software Science Sichuan Normal University Chengdu 610066 China Departmentof Mathematical Sciences Tsinghua University Beijing 100084 China. Department of Mathematical Sciences Tsinghua University Beijing 100084 China.

A formula on the complexity of the normal bases generated by prime Gauss period overfinite fields is presented in terms of cyclotomic ***,the authors determine explicitly thecomplexity of such normal bases and their dual bases in several cases where the related cyclotomicnumbers have been ***,the authors find several series of such normal bases withlow complexity.

关键词： Complexity cyclotomic number finite field Gauss period normal basis.

A New Criterion on k-Normal Elements over Finite Fields

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Chinese Annals of Mathematics,Series B 2020年第5期41卷 665-678页

作者： Aixian ZHANG keqin feng Department of Mathematical Sciences Xi’an University of TechnologyXi’an 710054China Department of Mathematical Sciences Tsinghua UniversityBeijing 100084China

The notion of normal elements for finite fields extension was generalized as k-normal elements by Huczynska et al.(2013).Several methods to construct k-normal elements were presented by Alizadah et al.(2016)and Huczynska et al.(2013),and the criteria on k-normal elements were given by Alizadah et al.(2016)and Antonio et al.(2018).In the paper by Huczynska,S.,Mullen,G.,Panario,*** Thomson,D.(2013),the number of k-normal elements for a fixed finite field extension was calculated and *** this paper the authors present a new criterion on k-normal elements by using idempotents and show some *** criterion was given for usual normal elements before by Zhang et al.(2015).

关键词： Normal basis Finite field Idempotent Linearized polynomial Gauss

Inhomogeneous Quantum Codes (III):The Asymmetric Case

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Chinese Annals of Mathematics,Series B 2014年第2期35卷 271-284页

作者： Weiyang WANG keqin feng School of Mathematical Sciences Capital Normal University Department of Mathematical Sciences Tsinghua University

The stabilizer(additive)method and non-additive method for constructing asymmetric quantum codes have been *** this paper,these methods are generalized to inhomogeneous quantum codes.

关键词： Inhomogeneous quantum codes Mixed classical codes Asymmetricquantum codes

Inhomogeneous quantum codes （II）： non-additive case

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Frontiers of Mathematics in China 2012年第3期7卷 573-586页

作者： Weiyang WANG keqin feng School of Mathematical Sciences Capital Normal University Beijing 100048 China Department of Mathematical Sciences Tsinghua University Beijing100084 China

The quantum codes have been generalized to inhomogeneous case and the stabilizer construction has been established to get additive inhomogeneous quantum codes in [Sei. China Math., 2010, 53： 2501-2510]. In this paper, we generalize the known constructions to construct non-additive inhomogeneous quantum codes and get examples of good d-ary quantum codes.

关键词： Quantum code inhomogeneous code mixed code finite abeliangroup character quadratic generalized boolean function

New series of odd non-congruent numbers

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Science China Mathematics 2006年第11期49卷 1642-1654页

作者： feng keqin XUE Yan Department of Mathematical Science Tsinghua University Beijing 100084 China Department of Mathematical Science Tsinghua University Beijing 100084 China

We determine all square-free odd positive integers n such that the 2-Selmer groups Sn and (S)n of the elliptic curve En: y2 = x(x - n)(x - 2n) and its dual curve (E)n: y2 =x3 + 6nx2 + n2x have the smallest size: Sn = {1}, (S)n = {1, 2, n, 2n}. It is well known that for such integer n, the rank of group En(Q) of the rational points on En is zero so that n is a non-congruent number. In this way we obtain many new series of elliptic curves En with rank zero and such series of integers n are non-congruent numbers.

关键词： congruent number, elliptic curves, rank, 2-descent, odd graph.

Growth kinetics and mechanism of microarc oxidation coating on Ti-6Al-4V alloy in phosphate/silicate electrolyte

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International Journal of Minerals,Metallurgy and Materials 2022年第11期29卷 1991-1999页

作者： Dajun Zhai Tao Qiu Jun Shen keqin feng College of Materials Science and Engineering Chongqing UniversityChongqing 400044China School of Mechanical Engineering Sichuan UniversityChengdu 610065China

Microarc oxidation(MAO)is an effective surface treatment method for Ti alloys to allow their application in extreme ***,binary electrolytes consisting of different amounts of sodium phosphate and sodium silicate were designed for *** surface morphology,composition,and properties of MAO coatings on Ti-6Al-4V alloy treated in 0.10 mol/L electrolyte were investigated to reveal the effect of PO_(4)^(3-)and SiO_(3)^(2-)ray diffraction,and potentiodynamic *** results showed that PO_(4)^(3-)is beneficial for generating microarcs and forming pores within the coating,resulting in a thick but porous ***_(3)^(2-)eration of microarcs,resulting in a thin dense *** thickness,density,phases content,and polarization resistance of the MAO coatings are primarily affected by the intensity of microarcs for low SiO_(3)^(2-)ciently *** thickness of MAO coatings obtained in P/Si electrolytes shows a piecewise linear increase with increasing process time during the three stages of microarc ***_(3)^(2-)discharge,but slows down the growth of the coating formed in the next stage.

关键词： microarc oxidation titanium alloy phosphate silicate growth kinetics growth mechanism

On the p-Adic Complexity of the Ding-Helleseth-Martinsen Binary Sequences

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Chinese Journal of Electronics 2021年第1期30卷 64-71页

作者： JING Xiaoyan XU Zhefeng YANG Minghui feng keqin Research Center for Number Theory and Its Applications Northwest University State Key Laboratory of Information Security Institute of Information Engineering Chinese Academy of Sciences Department of Mathematical Sciences Tsinghua University

The purpose of this paper is to determine the p-adic complexity of the Ding-Helleseth-Martinsen（DHM） sequences with period N = 2 q, where q ≡ 5（mod 8） is a prime number. We firstly use the p-adic exponential valuation, cyclotomic numbers of order four,"Gauss periods" and "quadratic Gauss sums" on finite field Fq and valued in Z;to determine the p-adic complexity of the DHM sequences.

关键词： sequences number theory binary sequences Gauss periods adic exponential valuation polynomials Ding-Helleseth-Martinsen binary prime number DHM sequences computational complexity adic complexity cyclotomic numbers Ding-Helleseth-Martinsen sequences

Perfect State Transfer on Weighted Abelian Cayley Graphs

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Chinese Annals of Mathematics,Series B 2021年第4期42卷 625-642页

作者： Xiwang CAO keqin feng Ying-Ying TAN Department of Mathematics Nanjing University of Aeronautics and Astronau-ticsNanjing 210016China Department of Mathematical Sciences Tsinghua UniversityBeijing 100084China School of Mathematics and Physics Anhui Jianzhu UniversityHefei 230601China

Recently,there are extensive studies on perfect state transfer(PST for short)on graphs due to their significant applications in quantum information processing and quantum ***,there is not any general characterization of graphs that have PST in *** this paper,the authors present a depiction on weighted abelian Cayley graphs having *** give a unified approach to describe the periodicity and the existence of PST on some specific graphs.

关键词： Perfect state transfer Cayley graph Eigenvalues of a graph Weighted graph Random walk

The Gauss sums and Jacobi sums over Galois ring GR(p^2 , r)

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Science China Mathematics 2013年第7期56卷 1457-1465页

作者： LI Jin ZHU ShiXin feng keqin School of Computer and Information Hefei University of Technology School of Mathematics Hefei University of Technology Department of Mathematical Sciences Tsinghua University

Galois rings and exponential sums over Galois rings have many applications in algebraic combinatorics, coding theory and cryptography. In this paper, we present explicit description on the Gauss sums and Jacobi sums over Galois ring GR(p2 , r), and show that the values of these sums can be reduced to the Gauss sums and Jacobi sums over finite field Fpr for all non-trivial cases.

关键词： Galois rings Gauss sums Jacobi sums finite fields