Constructions of maximum distance separable symbol-pair codes using cyclic and constacyclic codes

Symbol-pair code is a new coding framework which is proposed to correct errors in the symbol-pair read channel. In particular, maximum distance separable (MDS) symbol-pair codes are a kind of symbol-pair codes with the best possible error-correction capability. Employing cyclic and constacyclic codes, we construct three new classes of MDS symbol-pair codes with minimum pair-distance five or six. Moreover, we find a necessary and sufficient condition which ensures a class of cyclic codes to be MDS symbol-pair codes. This condition is related to certain property of a special kind of linear fractional transformations. A detailed analysis on these linear fractional transformations leads to an algorithm, which produces many MDS symbol-pair codes with minimum pair-distance seven.     

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.     

Codes of Small Defect

The parameters of a linear code C over GF(q) are given by [n,k,d], where n denotes the length, k the dimension and d the minimum distance of C. The code C is called MDS, or maximum distance separable, if the minimum distance d meets the Singleton bound, i.e. d = n-k+1 Unfortunately, the parameters of an MDS code are severely limited by the size of the field. Thus we look for codes which have minimum distance close to the Singleton bound. Of particular interest is the class of almost MDS codes, i.e. codes for which d=n-k. We will present a condition on the minimum distance of a code to guarantee that the orthogonal code is an almost MDS code. This extends a result of Dodunekov and Landgev Dodunekov. Evaluation of the MacWilliams identities leads to a closed formula for the weight distribution which turns out to be completely determined for almost MDS codes up to one parameter. As a consequence we obtain surprising combinatorial relations in such codes. This leads, among other things, to an answer to a question of Assmus and Mattson 5 on the existence of self-dual [2d,d,d]-codes which have no code words of weight d+1. Actually there are more codes than Assmus and Mattson expected, but the examples which we know are related to the expected ones.     

Direct constructions of additive codes

A code is q^m‐ary q‐linear if its alphabet forms an m‐dimensional vector space over ??_q and the code is linear over ??_q. These additive codes form a natural generalization of linear codes. Our main results are direct constructions of certain families of additive codes. These comprise the additive generalization of the Kasami codes, an additive generalization of the Bose‐Bush construction of orthogonal arrays of strength 2 as well as a class of additive codes which are being used for deep space communication. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 207–216, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.20000     

Multidimensional Latin bitrades

A subset of the n-dimensional k-valued hypercube is a unitrade or united bitrade whenever the size of its intersections with the one-dimensional faces of the hypercube takes only the values 0 and 2. A unitrade is bipartite or Hamiltonian whenever the corresponding subgraph of the hypercube is bipartite or Hamiltonian. The pair of parts of a bipartite unitrade is an n-dimensional Latin bitrade. For the n-dimensional ternary hypercube we determine the number of distinct unitrades and obtain an exponential lower bound on the number of inequivalent Latin bitrades. We list all possible n-dimensional Latin bitrades of size less than 2 ⁿ⁺¹. A subset of the n-dimensional k-valued hypercube is a t-fold MDS code whenever the size of its intersection with each one-dimensional face of the hypercube is exactly t. The symmetric difference of two single MDS codes is a bipartite unitrade. Each component of the corresponding Latin bitrade is a switching component of one of these MDS codes. We study the sizes of the components of MDS codes and the possibility of obtaining Latin bitrades of a size given from MDS codes. Furthermore, each MDS code is shown to embed in a Hamiltonian 2-fold MDS code.     

Local tensor valuations on convex polytopes

Local versions of the Minkowski tensors of convex bodies in $n$ -dimensional Euclidean space are introduced. An extension of Hadwiger’s characterization theorem for the intrinsic volumes, due to Alesker, states that the continuous, isometry covariant valuations on the space of convex bodies with values in the vector space of symmetric $p$ -tensors are linear combinations of modified Minkowski tensors. We ask for a local analogue of this characterization, and we prove a classification result for local tensor valuations on polytopes, without a continuity assumption.     

Isometries and additive mapping on the unit spheres of normed spaces

In this paper, we investigate isometric extension problem in general normed space. We prove that an isometry between spheres can be extended to a linear isometry between the spaces if and only if the natural positive homogeneous extension is additive on spheres. Moreover, this conclusion still holds provided that the additivity holds on a restricted domain of spheres.     

一类具有双恢复集的局部恢复码的构造

假设C是有限域Fq上的[n,k]线性码,如果码字的每个坐标是其它至多r个坐标的函数,称C是(n,k,r)局部恢复码,这里r是较小的数.在分布式存储系统中,具有多个恢复集的局部恢复码使得数据在系统中更具实际意义,因为它可以避免热数据的频繁访问.引入代数函数域、特别是Hermite函数域去构造局部恢复码,这类局部恢复码具有双恢复集,并且码长可以突破字符集的大小的限制.结果表明,此构造方法得出的最小距离下界明显地改进了Alexander Barg的最小距离的下界.     

Characteristics of invariant weights related to code equivalence over rings

The Equivalence Theorem states that, for a given weight on an alphabet, every isometry between linear codes extends to a monomial transformation of the entire space. This theorem has been proved for several weights and alphabets, including the original MacWilliams’ Equivalence Theorem for the Hamming weight on codes over finite fields. The question remains: What conditions must a weight satisfy so that the Extension Theorem will hold? In this paper we provide an algebraic framework for determining such conditions, generalising the approach taken in Greferath and Honold (Proceedings of the Tenth International Workshop in Algebraic and Combinatorial Coding Theory (ACCT-10), pp. 106–111. Zvenigorod, Russia, 2006).     

A complete characterization of irreducible cyclic orbit codes and their Plücker embedding

Constant dimension codes are subsets of the finite Grassmann variety. The study of these codes is a central topic in random linear network coding theory. Orbit codes represent a subclass of constant dimension codes. They are defined as orbits of a subgroup of the general linear group on the Grassmannian. This paper gives a complete characterization of orbit codes that are generated by an irreducible cyclic group, i.e. a group having one generator that has no non-trivial invariant subspace. We show how some of the basic properties of these codes, the cardinality and the minimum distance, can be derived using the isomorphism of the vector space and the extension field. Furthermore, we investigate the Plücker embedding of these codes and show how the orbit structure is preserved in the embedding.     

Extending MDS Codes

A q-ary (n, k)-MDS code, linear or not, satisfies n ≤ q + k − 1. A code meeting this bound is said to have maximum length. Using purely combinatorial methods we show that an MDS code with n = q + k − 2 can be uniquely extended to a maximum length code if and only if q is even. This result is best possible in the sense that there is, for example, a non-extendable 4-ary (5, 4)-MDS code. It may be that the proof of our result is as interesting as the result itself. We provide a simple necessary and sufficient condition for code extendability. In future work, this condition might be suitably modified to give an extendability condition for arbitrary (shorter) MDS codes.Received December 1, 2003     

New uniqueness proofs for the (5, 8, 24), (5, 6, 12) and related Steiner systems

New elementary proofs of the uniqueness of certain Steiner systems using coding theory are presented. In the process some of the codes involved are shown to be unique.The uniqueness proof for the (5, 8, 24) Steiner system is due to John Conway. The blocks of the system are used to generate a length 24 binary code. Any two such codes are then shown to be equivalent up to a permutation of the coordinates. This code turns out to be the extended Golay code.In the uniqueness proof for the (4, 7, 23) system, the blocks generate a length 23 code which is extended to a length 24 code. The minimum weight vectors of this larger code hold a (5, 8, 24) Steiner system. This result together with the previous one completes the proof. At this point it is also possible to conclude that the codes involved are unique and hence equivalent to the binary perfect Golay code and its extension.Continuing with the uniqueness result for the (3, 6, 22) Steiner system, the blocks generate a length 22 code which is extended to the same length 24 code by the addition of two coordinates and one additional vector. This extension ultimately requires the computation of the coset weight distribution of the length 22 code, a result heretofore unknown. The complete coset weight distribution for a specific (22, 11, 6) self-dual code is computed using the CAMAC computer system.The (5, 6, 12) and (4, 5, 11) Steiner systems are treated differently. It is shown that each system is completely determined by the choice of six blocks which may be assumed to lie in any such design. These six blocks in fact form a basis for length 12 (and 11) ternary codes corresponding to the two systems and may be generated by an algorithm independent of the designs. This algorithm is presented and the minimum weight vectors of the resulting codes, the perfect ternary Golay code and its extension, are calculated by the CAMAC system.     

赋β-范空间中单位球面间的等距算子的线性延拓

本文得到了等距映射的线性延拓的一般结果:设E,F是赋范(或β-严格凸赋β-范)线性空间,若V_0:S_1(E)→S_1(F)是等距,且对任意的x,y∈S_1(E),有‖V_0x-|(?)|V_0y‖≤‖x-|(?)|y‖,(?)∈R,则V_0必可延拓到全空间上等距算子(或线性等距算子)。特别,当E,F是赋范线性空间,V_0是满射或F为严格凸空间时,则V_0必可延拓为全空间的线性等距算子,从而推广了文[3～5]中的相应结果。     

Generalized Gabidulin codes over fields of any characteristic

We generalize Gabidulin codes to a large family of fields, non necessarily finite, possibly with characteristic zero. We consider a general field extension and any automorphism in the Galois group of the extension. This setting enables one to give several definitions of metrics related to the rank-metric, yet potentially different. We provide sufficient conditions on the given automorphism to ensure that the associated rank metrics are indeed all equal and proper, in coherence with the usual definition from linearized polynomials over finite fields. Under these conditions, we generalize the notion of Gabidulin codes. We also present an algorithm for decoding errors and erasures, whose complexity is given in terms of arithmetic operations. Over infinite fields the notion of code alphabet is essential, and more issues appear that in the finite field case. We first focus on codes over integer rings and study their associated decoding problem. But even if the code alphabet is small, we have to deal with the growth of intermediate values. A classical solution to this problem is to perform the computations modulo a prime ideal. For this, we need study the reduction of generalized Gabidulin codes modulo an ideal. We show that the codes obtained by reduction are the classical Gabidulin codes over finite fields. As a consequence, under some conditions, decoding generalized Gabidulin codes over integer rings can be reduced to decoding Gabidulin codes over a finite field.     

单位球面上的等距算子的延拓

本文考虑一般的Banach空间上的等距延拓问题,利用赋范集的概念给出了一些充分条件,使得单位球面间的满等距算子可以延拓为全空间上的线性等距算子。     

Scalar-flat K?hler orbifolds via quaternionic-complex reduction

We prove that any asymptotically locally Euclidean scalar-flat K?hler 4-orbifold whose isometry group contains a 2-torus is isometric, up to an orbifold covering, to a quaternionic-complex quotient of a k-dimensional quaternionic vector space by a (k−1)-torus. In order to do so, we first prove that any compact anti-self-dual 4-orbifold with positive Euler characteristic whose isometry group contains a 2-torus is conformally equivalent, up to an orbifold covering, to a quaternionic quotient of k-dimensional quaternionic projective space by a (k − 1)-torus.     

Maximum Distance Separable Codes in the ρ Metric over Arbitrary Alphabets

We give a bound for codes over an arbitrary alphabet in a non-Hamming metric and define MDS codes as codes meeting this bound. We show that MDS codes are precisely those codes that are uniformly distributed and show that their weight enumerators based on this metric are uniquely determined.     

Roos bound for skew cyclic codes in Hamming and rank metric

In this paper, a Roos like bound on the minimum distance for skew cyclic codes over a general field is provided. The result holds in the Hamming metric and in the rank metric. The proofs involve arithmetic properties of skew polynomials and an analysis of the rank of parity-check matrices. For the rank metric case, a way to arithmetically construct codes with a prescribed minimum rank distance, using the skew Roos bound, is also given. Moreover, some examples of MDS codes and MRD codes over finite fields are built, using the skew Roos bound.