Special solution of the （2＋1）-dimensional Sawada Kotera equation is decomposed into three （0＋1）- dimensional Bargmann flows. They are straightened out on the Jacobi variety of the associated hyperelliptic curve....Special solution of the （2＋1）-dimensional Sawada Kotera equation is decomposed into three （0＋1）- dimensional Bargmann flows. They are straightened out on the Jacobi variety of the associated hyperelliptic curve. Explicit algebraic-geometric solution is obtained on the basis of a deeper understanding of the KdV hierarchy.展开更多
A type of recurring relations on syndrome series is presnted. After some important theorems are established, an algorithm for computing a minimal polynomial set is presented. Finally an algorithm for determining the u...A type of recurring relations on syndrome series is presnted. After some important theorems are established, an algorithm for computing a minimal polynomial set is presented. Finally an algorithm for determining the unknown syndromes with incorporating α majority scheme is presented.展开更多
For a class of algebraic-geometric codes, a type of recurring relation is introduced on the syndrome sequence of an error vector. Then, a new majority voting scheme is developed. By applying the generalized Berlekamp-...For a class of algebraic-geometric codes, a type of recurring relation is introduced on the syndrome sequence of an error vector. Then, a new majority voting scheme is developed. By applying the generalized Berlekamp-Massey algorithm, and incorporating the majority voting scheme, an efficient decoding algorithm up to half the Feng-Rao bound is developed for a class of algebraic-geometric codes, the complexity of which is O(γο1n2), where n is the code length, and γ is the genus of curve. On different algebraic curves, the complexity of the algorithm can be lowered by choosing base functions suitably. For example, on Hermitian curves the complexity is O( n7/3 ).展开更多
Generally speaking, the parameters of algebraic-geometric codes are better, but there is still not a good decoding algorithm to get these codes go into practice. This note shows an algorithm with less computation. Let...Generally speaking, the parameters of algebraic-geometric codes are better, but there is still not a good decoding algorithm to get these codes go into practice. This note shows an algorithm with less computation. Let X be a smooth projective algebraic curve over F<sub>q</sub> which is a finite field with q elements, and P<sub>1</sub>, …, P<sub>n</sub> be n F<sub>q</sub>-rational points of X. Consider two divisors D, G:展开更多
A new Lax matrix is introduced for the integrable symplectic map (ISM), and the non-dynamical(i. e. constant) r-matrix of ISM is obtained. Moreover, an effective approach is systematically presented to construct the e...A new Lax matrix is introduced for the integrable symplectic map (ISM), and the non-dynamical(i. e. constant) r-matrix of ISM is obtained. Moreover, an effective approach is systematically presented to construct the explicit solution(here, the explicit solution means algebraic-geometric solution expressed by the Riemann-Theta function) of a soliton system or nonlinear evolution equation from Lax matrix, r-matrix, and the theory of nonlinearization through taking the Toda lattice as an example. The given algebraic-geometric solution of the Toda lattice is almost-periodic and includes the periodic and finite-band solution.展开更多
In this paper,the quantum error-correcting codes are generalized to the inhomogenous quantumstate space Cq1Cq2···Cqn,where qi(1 i n)are arbitrary positive integers.By attaching an abelian group Ai of or...In this paper,the quantum error-correcting codes are generalized to the inhomogenous quantumstate space Cq1Cq2···Cqn,where qi(1 i n)are arbitrary positive integers.By attaching an abelian group Ai of order qi to the space Cqi(1 i n),we present the stabilizer construction of such inhomogenous quantum codes,called additive quantum codes,in term of the character theory of the abelian group A=A1⊕A2⊕···⊕An.As usual case,such construction opens a way to get inhomogenous quantum codes from the classical mixed linear codes.We also present Singleton bound for inhomogenous additive quantum codes and show several quantum codes to meet such bound by using classical mixed algebraic-geometric codes.展开更多
In Ref.[2] the author dealt with the Main conjecture on geometric codes andproved the correctness of the conjecture for codes arising from curves with genus 1 or2 when the cardinal of the ground field is large enough....In Ref.[2] the author dealt with the Main conjecture on geometric codes andproved the correctness of the conjecture for codes arising from curves with genus 1 or2 when the cardinal of the ground field is large enough. In this note, the conjecturefor codes from hyperelliptic curves is attacked.展开更多
Propagation criteria and resiliency of vectorial Boolean functions are important for cryptographic purpose(see [1-4,7,8,10,11,16]).Kurosawa,Stoh [8] and Carlet [1] gave a construction of Boolean functions satisfying P...Propagation criteria and resiliency of vectorial Boolean functions are important for cryptographic purpose(see [1-4,7,8,10,11,16]).Kurosawa,Stoh [8] and Carlet [1] gave a construction of Boolean functions satisfying PC(l) of order k from binary linear or nonlinear codes.In this paper,the algebraic-geometric codes over GF(2m) are used to modify the Carlet and Kurosawa-Satoh's construction for giving vectorial resilient Boolean functions satisfying PC(l) of order k criterion.This new construction is compared with previously known results.展开更多
基金The project supported by the Special Funds for Major State Basic Research Project under Grant No.G2000077301
文摘Special solution of the （2＋1）-dimensional Sawada Kotera equation is decomposed into three （0＋1）- dimensional Bargmann flows. They are straightened out on the Jacobi variety of the associated hyperelliptic curve. Explicit algebraic-geometric solution is obtained on the basis of a deeper understanding of the KdV hierarchy.
文摘A type of recurring relations on syndrome series is presnted. After some important theorems are established, an algorithm for computing a minimal polynomial set is presented. Finally an algorithm for determining the unknown syndromes with incorporating α majority scheme is presented.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 69673025 and 69673016).
文摘For a class of algebraic-geometric codes, a type of recurring relation is introduced on the syndrome sequence of an error vector. Then, a new majority voting scheme is developed. By applying the generalized Berlekamp-Massey algorithm, and incorporating the majority voting scheme, an efficient decoding algorithm up to half the Feng-Rao bound is developed for a class of algebraic-geometric codes, the complexity of which is O(γο1n2), where n is the code length, and γ is the genus of curve. On different algebraic curves, the complexity of the algorithm can be lowered by choosing base functions suitably. For example, on Hermitian curves the complexity is O( n7/3 ).
文摘Generally speaking, the parameters of algebraic-geometric codes are better, but there is still not a good decoding algorithm to get these codes go into practice. This note shows an algorithm with less computation. Let X be a smooth projective algebraic curve over F<sub>q</sub> which is a finite field with q elements, and P<sub>1</sub>, …, P<sub>n</sub> be n F<sub>q</sub>-rational points of X. Consider two divisors D, G:
文摘A new Lax matrix is introduced for the integrable symplectic map (ISM), and the non-dynamical(i. e. constant) r-matrix of ISM is obtained. Moreover, an effective approach is systematically presented to construct the explicit solution(here, the explicit solution means algebraic-geometric solution expressed by the Riemann-Theta function) of a soliton system or nonlinear evolution equation from Lax matrix, r-matrix, and the theory of nonlinearization through taking the Toda lattice as an example. The given algebraic-geometric solution of the Toda lattice is almost-periodic and includes the periodic and finite-band solution.
基金supported by National Natural Science Foundation of China(Grant No.10990011)
文摘In this paper,the quantum error-correcting codes are generalized to the inhomogenous quantumstate space Cq1Cq2···Cqn,where qi(1 i n)are arbitrary positive integers.By attaching an abelian group Ai of order qi to the space Cqi(1 i n),we present the stabilizer construction of such inhomogenous quantum codes,called additive quantum codes,in term of the character theory of the abelian group A=A1⊕A2⊕···⊕An.As usual case,such construction opens a way to get inhomogenous quantum codes from the classical mixed linear codes.We also present Singleton bound for inhomogenous additive quantum codes and show several quantum codes to meet such bound by using classical mixed algebraic-geometric codes.
文摘In Ref.[2] the author dealt with the Main conjecture on geometric codes andproved the correctness of the conjecture for codes arising from curves with genus 1 or2 when the cardinal of the ground field is large enough. In this note, the conjecturefor codes from hyperelliptic curves is attacked.
基金Project supported by the National Natural Science Foundation of China (No.10871068)the joint grant of the Danish National Research Foundation and the National Natural Science Foundation of China and the Shanghai Leading Academic Discipline Project (No.S30504)
文摘Propagation criteria and resiliency of vectorial Boolean functions are important for cryptographic purpose(see [1-4,7,8,10,11,16]).Kurosawa,Stoh [8] and Carlet [1] gave a construction of Boolean functions satisfying PC(l) of order k from binary linear or nonlinear codes.In this paper,the algebraic-geometric codes over GF(2m) are used to modify the Carlet and Kurosawa-Satoh's construction for giving vectorial resilient Boolean functions satisfying PC(l) of order k criterion.This new construction is compared with previously known results.
