Inhomogenous quantum codes (Ⅰ):additive case

In this paper,the quantum error-correcting codes are generalized to the inhomogenous quantumstate space Cq1  Cq2 ··· 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.     

New Quantum MDS Code from Constacyclic Codes

In recent years, there have been intensive activities in the area of constructing quantum maximum distance separable(MDS for short) codes from constacyclic MDS codes through the Hermitian construction. In this paper, a new class of quantum MDS code is constructed, which extends the result of [Theorems 3.14–3.15, Kai, X., Zhu, S., and Li,P., IEEE Trans. on Inf. Theory, 60(4), 2014, 2080–2086], in the sense that our quantum MDS code has bigger minimum distance.     

Analysis of a moving collocation method for one-dimensional partial differential equations

A moving collocation method has been shown to be very efficient for the adaptive solution of second- and fourth-order time-dependent partial differential equations and forms the basis for the two robust codes MOVCOL and MOVCOL4.In this paper,the relations between the method and the traditional collocation and finite volume methods are investigated.It is shown that the moving collocation method inherits desirable properties of both methods: the ease of implementation and high-order convergence of the traditional collocation method and the mass conservation of the finite volume method.Convergence of the method in the maximum norm is proven for general linear two-point boundary value problems.Numerical results are given to demonstrate the convergence order of the method.     

A successive approximation method for quantum separability

Determining whether a quantum state is separable or inseparable （entangled） is a problem of fundamental importance in quantum science and has attracted much attention since its first recognition by Einstein, Podolsky and Rosen [Phys. Rev., 1935, 47： 777] and SchrSdinger [Naturwissenschaften, 1935, 23： 807-812, 823-828, 844-849]. In this paper, we propose a successive approximation method （SAM） for this problem, which approximates a given quantum state by a so-called separable state： if the given states is separable, this method finds its rank-one components and the associated weights; otherwise, this method finds the distance between the given state to the set of separable states, which gives information about the degree of entanglement in the system. The key task per iteration is to find a feasible descent direction, which is equivalent to finding the largest M-eigenvalue of a fourth-order tensor. We give a direct method for this problem when the dimension of the tensor is 2 and a heuristic cross-hill method for cases of high dimension. Some numerical results and experiences are presented.     

CYCLIC CODES OVER FORMAL POWER SERIES RINGS

In this article, cyclic codes and negacyclic codes over formal power series rings are studied. The structure of cyclic codes over this class of rings is given, and the relationship between these codes and cyclic codes over finite chain rings is obtained. Using an isomorphism between cyclic and negacyclic codes over formal power series rings, the structure of negacyclic codes over the formal power series rings is obtained.     

Self-dual Co des with Symplectic Inner Pro duct

In this paper, we discuss a kind of Hermitian inner product — symplectic inner product, which is different from the original inner product — Euclidean inner product. According to the definition of symplectic inner product, the codes under the symplectic inner product have better properties than those under the general Hermitian inner product. Here we present the necessary and sufficient condition for judging whether a linear code C over Fp with a generator matrix in the standard form is a symplectic self-dual code. In addition, we give a method for constructing a new symplectic self-dual codes over Fp, which is simpler than others.     

SELF-DUAL PERMUTATION CODES OVER FORMAL POWER SERIES RINGS AND FINITE PRINCIPAL IDEAL RINGS

In this paper, we study self-dual permutation codes over formal power series rings and finite principal ideal rings. We first give some results on the torsion codes associated with the linear codes over formal power series rings. These results allow for obtaining some conditions for non-existence of self-dual permutation codes over formal power series rings. Finally, we describe self-dual permutation codes over finite principal ideal rings by examining permutation codes over their component chain rings.     

Type Ⅱ codes over finite rings

In this paper,we generalize the concept of Type Ⅱ codes to arbitrary finite rings. We focus on Type Ⅱ codes over finite chain rings and use the Chinese Remainder Theorem on these codes to study Type Ⅱ codes over principal ideal rings.     

The Elements in Crystal Bases Corresponding to Exceptional Modules

According to the Ringel-Green theorem, the generic composition algebra of the Hall algebra provides a realization of the positive part of the quantum group. Furthermore, its Drinfeld double can be identified with the whole quantum group, in which the BGP- reflection functors coincide with Lusztig＇s symmetries. It is first asserted that the elements corresponding to exceptional modules lie in the integral generic composition algebra, hence in the integral form of the quantum group. Then it is proved that these elements lie in the crystal basis up to a sign. Eventually, it is shown that the sign can be removed by the geometric method. The results hold for any type of Cartan datum.     

环Fq+uFq+vFq上的循环码和量子码

本文研究了环R=F_q+uF_q+vF_q(u²=u,v²=v,uv=vu=0)上的循环码构造量子码的方法.利用环R上循环码的分解与生成多项式,给出了R上一个循环码可以构造量子码的一个充要条件.作为这类循环码的应用,得到了新的非二元量子码.     

Construction of some new quantum MDS codes

Quantum maximum-distance-separable (MDS) codes are an important class of quantum codes. In this paper, we mainly apply a new method of classical Hermitian self-orthogonal codes to construct three classes of new quantum MDS codes, and these quantum MDS codes provide large minimum distance.     

New quantum MDS codes with large minimum distance and short length from generalized Reed–Solomon codes

Quantum maximum-distance-separable (MDS) codes are an important class of quantum codes. In this paper we mainly use classical Hermitian self-orthogonal generalized Reed–Solomon codes to construct three classes of new quantum MDS codes. Further, these quantum MDS codes have large minimum distance and short length.     

Nonbinary quantum error-correcting codes from algebraic curves

We give a new exposition and proof of a generalized CSS construction for nonbinary quantum error-correcting codes. Using this we construct nonbinary quantum stabilizer codes with various lengths, dimensions, and minimum distances from algebraic curves. We also give asymptotically good nonbinary quantum codes from a Garcia–Stichtenoth tower of function fields which are constructible in polynomial time.     

量子纠错码的等价和保距同构

研究了量子纠错码的等价性和保距同构,推广了Bogart等人的一些概念,并给出若干基本定理,这些定理对进一步研究量子码的等价性和保距同构是非常有用的.在此基础上构造出一个反例,证明了在量子情形下,MacWmiams的一个重要定理不成立.     

New quantum codes from evaluation and matrix-product codes

Stabilizer codes obtained via the CSS code construction and the Steane's enlargement of subfield-subcodes and matrix-product codes coming from generalized Reed–Muller, hyperbolic and affine variety codes are studied. Stabilizer codes with good quantum parameters are supplied; in particular, some binary codes of lengths 127 and 128 improve the parameters of the codes in http://www.codetables.de. Moreover, non-binary codes are presented either with parameters better than or equal to the quantum codes obtained from BCH codes by La Guardia or with lengths that cannot be reached by them.     

Linear transformations on codes

This paper studies and classifies linear transformations that connect Hamming distances of codes. These include irreducible linear transformations and their concatenations. Their effect on the Hamming weights of codewords is investigated. Both linear and non-linear codes over fields are considered. We construct optimal linear codes and a family of pure binary quantum codes using these transformations.     

Fully Ramified Characters and Clifford Codes

In this article we study relations between groups and quantum error correcting codes. Groups of central type are used to construct quantum error correcting codes. Both stabilizer and Clifford codes can be derived from a construction involving this kind of groups. A more general construction of Clifford codes will be given and their correcting properties will be examined using group theoretical techniques.     

Constacyclic codes over finite commutative semi-simple rings

Finite commutative semi-simple rings are direct sum of finite fields. In this study, we investigate the algebraic structure of λ-constacyclic codes over such finite semi-simple rings. Among others, necessary and sufficient conditions for the existence of self-dual, LCD, and Hermitian dual-containing λ-constacyclic codes over finite semi-simple rings are provided. Using the CSS and Hermitian constructions, quantum MDS codes over finite semi-simple rings are constructed.