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 o...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.展开更多
In this paper, we discuss the relation between the partial sums of Jacobi serier on an elliptic region and the corresponding partial sums of Fourier series. From this we derive a precise approximation formula by the p...In this paper, we discuss the relation between the partial sums of Jacobi serier on an elliptic region and the corresponding partial sums of Fourier series. From this we derive a precise approximation formula by the partial sums of Jacobi series on an elliptic region.展开更多
In this paper, explicit determination of the cyclotomic numbers of order l and 2l, for odd prime l ≡ 3 (mod 4), over finite field Fq in the index 2 case are obtained, utilizing the explicit formulas on the correspond...In this paper, explicit determination of the cyclotomic numbers of order l and 2l, for odd prime l ≡ 3 (mod 4), over finite field Fq in the index 2 case are obtained, utilizing the explicit formulas on the corresponding Gauss sums. The main results in this paper are related with the number of rational points of certain elliptic curve, called "Legendre curve", and the properties and value distribution of such number are also presented.展开更多
Let Fqbe the finite field,q=p^(k),with p being a prime and k being a positive integer.Let F_(q)^(*)be the multiplicative group of Fq,that is F_(q)^(*)=F_(q){0}.In this paper,by using the Jacobi sums and an analog of H...Let Fqbe the finite field,q=p^(k),with p being a prime and k being a positive integer.Let F_(q)^(*)be the multiplicative group of Fq,that is F_(q)^(*)=F_(q){0}.In this paper,by using the Jacobi sums and an analog of Hasse-Davenport theorem,an explicit formula for the number of solutions of cubic diagonal equation x_(1)^(3)+x_(2)^(3)+…+x_(n)^(3)=c over Fqis given,where c∈F_(q)^(*)and p≡1(mod 3).This extends earlier results.展开更多
Several new series of approximately mutually unbiased bases are constructed by using Gauss sums and Jacobi sums over Galois rings GR(p2, r), and the tensor method.
基金supported by National Natural Science Foundation of China(Grant Nos.60973125 and 10990011)Science and Technology on Information Assurance Lab(Grant No.KJ-12-01)the Tsinghua National Lab for Information Science and Technology
文摘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.
文摘In this paper, we discuss the relation between the partial sums of Jacobi serier on an elliptic region and the corresponding partial sums of Fourier series. From this we derive a precise approximation formula by the partial sums of Jacobi series on an elliptic region.
基金supported by National Natural Science Foundation of China(Grant Nos.10990011,11001145 and 61170289)the Science and Technology on Information Assurance Laboratory Foundation(Grant No.KJ-12-01)
文摘In this paper, explicit determination of the cyclotomic numbers of order l and 2l, for odd prime l ≡ 3 (mod 4), over finite field Fq in the index 2 case are obtained, utilizing the explicit formulas on the corresponding Gauss sums. The main results in this paper are related with the number of rational points of certain elliptic curve, called "Legendre curve", and the properties and value distribution of such number are also presented.
基金Supported by the Natural Science Foundation of Henan Province(232300420123)the National Natural Science Foundation of China(12026224)the Research Center of Mathematics and Applied Mathematics,Nanyang Institute of Technology。
文摘Let Fqbe the finite field,q=p^(k),with p being a prime and k being a positive integer.Let F_(q)^(*)be the multiplicative group of Fq,that is F_(q)^(*)=F_(q){0}.In this paper,by using the Jacobi sums and an analog of Hasse-Davenport theorem,an explicit formula for the number of solutions of cubic diagonal equation x_(1)^(3)+x_(2)^(3)+…+x_(n)^(3)=c over Fqis given,where c∈F_(q)^(*)and p≡1(mod 3).This extends earlier results.
基金supported by the Natural Science Foundation of China under Grant No.61370089the Tsinghua National Laboratory for Information Science and Technology+1 种基金by the Fundamental Research Funds for the Central Universities under Grant No.JZ2014HGBZ0349by Science and Technology on Information Assurance Lab.KJ-12-01
文摘Several new series of approximately mutually unbiased bases are constructed by using Gauss sums and Jacobi sums over Galois rings GR(p2, r), and the tensor method.