Exponential Sums over Finite Fields

This is an expository paper on algebraic aspects of exponential sums over finite fields. This is a new direction. Various examples, results and open problems are presented along the way, with particular emphasis on Gauss periods, Kloosterman sums and one variable exponential sums. One main tool is the applications of various p-adic methods. For this reason, the author has also included a brief exposition of certain p-adic estimates of exponential sums. The material is based on the lectures given at the 2020 online number theory summer school held at Xiamen University. Notes were taken by Shaoshi Chen and Ruichen Xu.     

Optimal p-Ary Codes from One-Weight and Two-Weight Codes over IF_p+ vIF_p

This paper is devoted to determining the structures and properties of one-Lee weight codes and two-Lee weight projective codes Ck1,k2,k3 over p IF+ v IFp with type p2k1pk2pk3. The authors introduce a distance-preserving Gray map from( IFp + v IFp)nto2np. By the Gray map, the authors construct a family of optimal one-Hamming weight p-ary linear codes from one-Lee weight codes over IFp+ v IFp, which attain the Plotkin bound and the Griesmer bound. The authors also obtain a class of optimal p-ary linear codes from two-Lee weight projective codes over IFp + vIFp, which meet the Griesmer bound.     

基于LDPC码的WCDMA下行链路性能仿真

LDPC码是一类靠近Shannon限的高效信道编码，在AWGN信道下具有非常优越的性能。在讨论了LDPC码的基本原理之后，构建了基于LDPC码的WCDMA下行链路，改进了LDPC码的译码过程。仿真结果表明，在高速传输速率下（384kbit／s），不规则LDPC码具有优于Turbo码的性能，这对于LDPC码应用到复杂无线通信系统中具有重要意义。     

Construction of One-Gray Weight Codes and Two-Gray Weight Codes over Z_4+uZ_4

This paper firstly gives some necessary conditions on one-Gray weight linear codes. And then we use these results to construct several classes of one-Gray weight linear codes over Z_4 +uZ_4(u~2=u) with type 16~(k_1)8~(k_2)8~(k_3)4~(k_4)4~(k_5)4~(k_6)2~(k_7)2~(k_8) based on a distance-preserving Gray map from(Z4 + u Z4)n to Z2n4. Secondly, the authors use the similar approach to do works on two-Gray(projective) weight linear codes. Finally, some examples are given to illustrate the construction methods.     

On the Gray Images of Some Constacyclic Codes over F_p+uF_p+u~2F_p

Constacyclic codes are an important class of linear codes in coding theory.Many optimal linear codes are directly derived from constacyclic codes.In this paper,a new Gray map between codes over F_p+uF_p+u~2F_p and codes over Fp is defined,where p is an odd prime.By means of this map,it is shown that the Gray image of a linear(1+u+u~2)-constacyclic code over F_p+uF_p+u~2F_p of length n is a repeated-root cyclic code over F_p of length pn.Furthermore,some examples of optimal linear cyclic codes over F_3 from(1+u+u~2)-constacyclic codes over F_3+uF_3+u~2F_3 are given.     

THE HAMMING DISTANCES OF NEGACYCLIC CODES OF LENGTH 2^s OVER GR （2^α, m）

The Hamming distances of all negacyclic codes of length 28 over the Gaiois ring GR （2^α, m） are given. In particular, the Lee distances of all negacyclic codes over Z4 of length 2^s are obtained. The Gray images of such negacyclic codes over Z4 are also determined under the Gray map.     

A Class of Constacyclic Codes over Ring R+vR

By constructing a Gray map,a class of constacyclic codes over ring=■+vR is studied.Using cyclic codes and negacyclic codes of length p~s over ring R,the structure of(1-2v)-constacyclic codes and dual codes of length p~s over ring are given,the Gray images of(1-2v)-constacyclic codes in a particular case are also studied.It is shown that linear codes of length p~s over ring ■ are(1-2v)-constacyclic codes if and only if their Gray images are distance-invariant cyclic codes of length2p~s over ring R.     

Construction of One-Lee Weight and Two-Lee Weight Codes over IF_p+vIF_p

This paper is devoted to the construction of one-Lee weight codes and two-Lee weight codes over IF_p+vIF_p(v~2=v) with type p~(2 k_1)p~(k2)p~(k3) based on two different distance-preserving Gray maps from((IF_p+vIF_p)~n, Lee weight) to(IF_p~(2 n), Hamming weight), where p is a prime. Moreover, the authors prove that the obtained two-Lee weight codes are projective only when p=2.     

On Some Computational Problems in Local Fields

Journal of Systems Science and Complexity - Lattices in Euclidean spaces are important research objects in geometric number theory, and they have important applications in many areas, such as...     

气象数据标量场的建模与可视化研究

气象标量场的可视化仍有着许多效率、交互性和准确性上的欠缺。针对这些问题,提出基于多层次体绘制技术的显示方法,对空间场数据进行处理,形成三维空间网格,通过标量值与颜色值、透明度之间的动态映射,实现气象标量数据的三维动态可视化效果。针对传统MC(Marching Cubes)算法存在的插值方法不适用于气象数据,连接方式具有二义性的问题,提出MC算法的改进方法,完善了MC算法在气象可视化领域的不足。实验证明,体绘制能够快速逼真地仿真出气象数据的动态变化效果,改进后的MC算法则能使结果更加精确。     

Ramp Scheme Based on CRT for Polynomial Ring over Finite Field

Chinese Reminder Theorem(CRT) for integers has been widely used to construct secret sharing schemes for different scenarios, but these schemes have lower information rates than that of Lagrange interpolation-based schemes. In ASIACRYPT 2018, Ning, et al. constructed a perfect(r, n)-threshold scheme based on CRT for polynomial ring over finite field, and the corresponding information rate is one which is the greatest case for a(r, n)-threshold scheme. However, for many practical purposes, the inf...     

Macwilliams Identities of Linear Codes over the Ring F_2+ uF_2+ vF_2

The Lee weight enumerators and the complete weight enumerators for the linear codes over ring R = F2 + u F2 + v F2 are defined and Gray map from Rnto F3n2 is constructed. By proving the fact that the Gray images of the self-dual codes over R are the self-dual codes over F2, and based on the Mac Williams identities for the Hamming weight enumerators of linear codes over F2, the Mac Williams identities for Lee weight enumerators of linear codes over R are given. Further, by introducing a special variable t, the Mac Williams identities for the complete weight enumerators of linear codes over R are obtained. Finally, an example which illustrates the correctness and function of the two Mac Williams identities is provided.     

AZ31镁合金快速铸轧过程热应力场有限元仿真研究

近年来变形镁合金得到了广泛的研究和应用,薄板快速铸轧技术为新型变形镁合金提供质量合格的板坯.为了研究铸轧工艺参数对AZ31镁合金薄板快速铸轧过程温度场和热应力场的影响,根据读技术在镁合金中应用的特点基于铸轧区板坯的对称性建立了纵截面1/2的二维几何模型;选择自由网格划分法和Smart-Sizing算法划分有限元网格;采用大型通用有限元分析软件ANSYS对镁合金快速铸轧过程中的铸坯温度场和热-应力场进行了仿真分析,并就不同工艺参数(接触界面换热系数、铸轧速度)对铸坯温度和应力的分布及其相变区的影响进行了研究.仿真结果增强了对镁合金快速铸轧过程相变区温度变化和热裂产生机制的理解,为快速铸轧工艺参数的优化提供了依据.     

Recent Results on Constructing Boolean Functions with (Potentially) Optimal Algebraic Immunity Based on Decompositions of Finite Fields

Boolean functions with optimal algebraic immunity (OAI functions) are important cryptographic primitives in the design of stream ciphers. During the past decade, a lot of work has been done on constructing such functions, among which mathematics, especially finite fields, play an important role. Notably, the approach based on decompositions of additive or multiplicative groups of finite fields turns out to be a very successful one in constructing OAI functions, where some original ideas are contributed by Tu and Deng (2012), Tang, et al. (2017), and Lou, et al. (2015). Motivated by their pioneering work, the authors and their collaborators have done a series of work, obtaining some more general constructions of OAI functions based on decompositions of finite fields. In this survey article, the authors review our work in this field in the past few years, illustrating the ideas for the step-by-step generalizations of previous constructions and recalling several new observations on a combinatorial conjecture on binary strings known as the Tu-Deng conjecture. In fact, the authors have obtained some variants or more general forms of Tu-Deng conjecture, and the optimal algebraic immunity of certain classes of functions we constructed is based on these conjectures.

     

On Basis and Pure Nash Equilibrium of Finite Pure Harmonic Games

This paper investigates the basis and pure Nash equilibrium of finite pure harmonic games (FPHGs) based on the vector space structure. First, a new criterion is proposed for the construction of pure harmonic subspace, based on which, a more concise basis is constructed for the pure harmonic subspace. Second, based on the new basis of FPHGs and auxiliary harmonic vector, a more easily verifiable criterion is presented for the existence of pure Nash equilibrium in basis FPHGs. Third, by constructing a pure Nash equilibrium cubic matrix, the verification of pure Nash equilibrium in three-player FPHGs is given.

     

瑞利衰落信道下SR-ARQ-AMS系统时延性能研究

针对瑞利衰落信道,对一种结合选择式自动请求重传(SR-ARQ)技术的自适应调制系统(SR-ARQ-AMS)及其传输原理进行了分析,构造了一个有限状态Markov链来描述其瑞利衰落信道.利用等效时延的思想,建立了发送端带有休假的M/G/1非空竭服务排队模型描述该自适应调制系统.考虑了经有限次传输未成功分组的服务时延对系统时延性能的影响,求得了该系统分组平均等待时延、分组平均时延和归一化最大吞吐量的解析式.通过数值模拟分析了各参数对系统时延性能的影响,进而得到提高系统传输速率的方法.仿真结果表明SR-ARQ-AMS在瑞利衰落信道下能获得更好的系统时延性能.     

On the Construction of Involutory MDS Matrices over F_(2~m)

This paper studies the problem of constructing lightweight involutory maximal distance separable(MDS) matrices. The authors find the exact lower bound of the XOR counts for 4 × 4 involutory MDS matrices over F_(2~4). Further, some new structures of 4 × 4 involutory MDS matrices over F_(2~m) are provided to construct involutory MDS matrices and the authors constructed the lightest4 × 4 involutory MDS matrices over F_(2~8) so far by using these structures.     

脉冲噪声衰落信道中正交空时分组码的误码率

从信噪比(SNR)观点对脉冲噪声环境下衰落信道中正交空时分组码(OSTBC)的理论误符号率(SER)进行了分析。基于STBC引入的标量等效矩阵,给出了STBC解码后每符号瞬时SNR的概率密度函数。假设信道为准静态平坦衰落,并将信道干扰噪声建模为ε-高斯混合分布。推导了Rayleigh、Rician及Nakagami-m三种衰落信道下采用MPSK和MQAM调制方式的OSTBCSER闭式表达式。该闭式解可以用于评估脉冲噪声环境中OSTBC的性能。此外,进行了蒙特卡洛仿真验证了分析的正确性。     

On the complexity of the normal bases via prime Gauss period over finite fields

A formula on the complexity of the normal bases generated by prime Gauss period over finite fields is presented in terms of cyclotomic numbers. Then, the authors determine explicitly the complexity of such normal bases and their dual bases in several cases where the related cyclotomic numbers have been calculated. Particularly, the authors ˉnd several series of such normal bases with low complexity. This research is supported by the National Fundamental Science Research Program 973 of China under Grant No. 2004 CB3180000 and the State Key Lab. (Information Security) of China.