In this paper,we use the elementary methods,the properties of Dirichlet character sums and the classical Gauss sums to study the estimation of the mean value of high-powers for a special character sum modulo a prime,a...In this paper,we use the elementary methods,the properties of Dirichlet character sums and the classical Gauss sums to study the estimation of the mean value of high-powers for a special character sum modulo a prime,and derive an exact computational formula.It can be conveniently programmed by the“Mathematica”software,by which we can get the exact results easily.展开更多
Sequences with nice pseudo-randomness play an important role in not only communication system but also cryptography system. Based on the Legendre-Sidelnikov sequence, a modified Legendre-Sidelnikov sequence was introd...Sequences with nice pseudo-randomness play an important role in not only communication system but also cryptography system. Based on the Legendre-Sidelnikov sequence, a modified Legendre-Sidelnikov sequence was introduced. The exact value of the autocorrelation function was derived by strict computation. According to the values of the autocorrelation functions of the two Legendre-Sidelnikov sequences, it is proven that both of them have perfect pseudo-randomness. Furthermore, a detailed comparison between autocorrelation functions of the two Legendre-Sidelnikov sequences was deduced. It indicates that no matter which parameters are chosen, the modified sequence has pseudo-randomness as good as the primitive sequence, which is of great significance for applications.展开更多
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...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.展开更多
Based on the quadratic exponential method, this paper constructs two types of generators over finite field Fq, the digital quadratic exponential generator and quadratic exponential pseudorandom vector generator. We in...Based on the quadratic exponential method, this paper constructs two types of generators over finite field Fq, the digital quadratic exponential generator and quadratic exponential pseudorandom vector generator. We investigate the distribution of the sequence generated by the generators, and present results about their one dimensional discrepancy. The proofs are based on the estimate of certain character sum over Fq. Ift is the least period of the sequence and t≥q^1/2＋2c, then the bound of the discrepancy is O（t^-1/4q^1/8＋τ logq） for any ε 〉 0. It shows that the sequence is asymptotically uniformly distributed.展开更多
Let q be a large prime, and χ the quadratic character modulo q. Let Φ be a self-dual Hecke-Maass cusp form for SL(3, Z), and uj a Hecke-Maass cusp form for Γ0(q) ■ SL(2, Z) with spectral parameter tj. We prove, fo...Let q be a large prime, and χ the quadratic character modulo q. Let Φ be a self-dual Hecke-Maass cusp form for SL(3, Z), and uj a Hecke-Maass cusp form for Γ0(q) ■ SL(2, Z) with spectral parameter tj. We prove, for the first time, some hybrid subconvexity bounds for the twisted L-functions on GL(3), such as L(1/2, Φ× uj ×χ) ■Φ,ε(q(1 + |tj |))3/2-θ+ε and L(1/2 + it, Φ×χ) ■Φ,ε(q(1 + |t|))3/4-θ/2+ε for any ε > 0, where θ = 1/23 is admissible. The proofs depend on the first moment of a family of L-functions in short intervals. In order to bound this moment, we first use the approximate functional equations, the Kuznetsov formula, and the Voronoi formula to transform it to a complicated summation, and then we apply different methods to estimate it, which give us strong bounds in different aspects. We also use the stationary phase method and the large sieve inequalities.展开更多
For a kind of L-function of the real quadratic number fields, we prove a Kronecker limit formula which generalized a result of Hecke. And taking an example we give an interesting identity on a fundamental unit of such...For a kind of L-function of the real quadratic number fields, we prove a Kronecker limit formula which generalized a result of Hecke. And taking an example we give an interesting identity on a fundamental unit of such a field.展开更多
Let p be an odd prime and （n/p） be the Legendre symbol.When p≡1（mod4）,it is easily seen that the numbers of quadratic residues in intervals T<sub>1</sub> =[1,（p-1）/2] and T2=[（p+1）/2,p] are equal....Let p be an odd prime and （n/p） be the Legendre symbol.When p≡1（mod4）,it is easily seen that the numbers of quadratic residues in intervals T<sub>1</sub> =[1,（p-1）/2] and T2=[（p+1）/2,p] are equal.In other words, the distribution of quadratic展开更多
文摘In this paper,we use the elementary methods,the properties of Dirichlet character sums and the classical Gauss sums to study the estimation of the mean value of high-powers for a special character sum modulo a prime,and derive an exact computational formula.It can be conveniently programmed by the“Mathematica”software,by which we can get the exact results easily.
基金supported by the National Natural Science Foundation of China (60833008)the Science and Technology on Communication Security Laboratory (9140C110201110C1102)the Fundamental Research Funds for the Central Universities (K5051270003, K50511010007)
文摘Sequences with nice pseudo-randomness play an important role in not only communication system but also cryptography system. Based on the Legendre-Sidelnikov sequence, a modified Legendre-Sidelnikov sequence was introduced. The exact value of the autocorrelation function was derived by strict computation. According to the values of the autocorrelation functions of the two Legendre-Sidelnikov sequences, it is proven that both of them have perfect pseudo-randomness. Furthermore, a detailed comparison between autocorrelation functions of the two Legendre-Sidelnikov sequences was deduced. It indicates that no matter which parameters are chosen, the modified sequence has pseudo-randomness as good as the primitive sequence, which is of great significance for applications.
文摘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.
基金Supported by the Special Fund of National Excellent Doctoral Dissertation (Grant 200060) and the National Natural Science Foundation of China (No.60373092).
文摘Based on the quadratic exponential method, this paper constructs two types of generators over finite field Fq, the digital quadratic exponential generator and quadratic exponential pseudorandom vector generator. We investigate the distribution of the sequence generated by the generators, and present results about their one dimensional discrepancy. The proofs are based on the estimate of certain character sum over Fq. Ift is the least period of the sequence and t≥q^1/2＋2c, then the bound of the discrepancy is O（t^-1/4q^1/8＋τ logq） for any ε 〉 0. It shows that the sequence is asymptotically uniformly distributed.
基金supported by National Natural Science Foundation of China(Grant No.11531008)the Ministry of Education of China(Grant No.IRT 16R43)。
文摘Let q be a large prime, and χ the quadratic character modulo q. Let Φ be a self-dual Hecke-Maass cusp form for SL(3, Z), and uj a Hecke-Maass cusp form for Γ0(q) ■ SL(2, Z) with spectral parameter tj. We prove, for the first time, some hybrid subconvexity bounds for the twisted L-functions on GL(3), such as L(1/2, Φ× uj ×χ) ■Φ,ε(q(1 + |tj |))3/2-θ+ε and L(1/2 + it, Φ×χ) ■Φ,ε(q(1 + |t|))3/4-θ/2+ε for any ε > 0, where θ = 1/23 is admissible. The proofs depend on the first moment of a family of L-functions in short intervals. In order to bound this moment, we first use the approximate functional equations, the Kuznetsov formula, and the Voronoi formula to transform it to a complicated summation, and then we apply different methods to estimate it, which give us strong bounds in different aspects. We also use the stationary phase method and the large sieve inequalities.
基金partly supported by the National Natural Science Foundation of China ( Grant No. 19531020) a Tian Yuan-Area-Grant.
文摘For a kind of L-function of the real quadratic number fields, we prove a Kronecker limit formula which generalized a result of Hecke. And taking an example we give an interesting identity on a fundamental unit of such a field.
文摘Let p be an odd prime and （n/p） be the Legendre symbol.When p≡1（mod4）,it is easily seen that the numbers of quadratic residues in intervals T<sub>1</sub> =[1,（p-1）/2] and T2=[（p+1）/2,p] are equal.In other words, the distribution of quadratic
