Curves in Projective Spaces and Almost MDS Codes

A k-track in PG(n,q) is a set of k points such that every n of them are in general position. Here, we construct a class of n-tracks arising from algebraic curves.     

平面曲线上几何MDS码的主猜想

本文用Ｌａｎｇ－Ｗｅｉｌ的一个经典结果证明了在一定维数限制下充分大域上平面代数曲线上ＭＤＳ码的主猜想成立。     

A New Approach to the Main Conjecture on Algebraic-Geometric MDS Codes

The Main Conjecture on MDS Codes statesthat for every linear [n, k] MDS code over q, if 1 <k < q, then n q+1,except when q is even and k=3 or k=q-1,in which cases n q +2. Recently, there has beenan attempt to prove the conjecture in the case of algebraic-geometriccodes. The method until now has been to reduce the conjectureto a statement about the arithmetic of the jacobian of the curve,and the conjecture has been successfully proven in this way forelliptic and hyperelliptic curves. We present a new approachto the problem, which depends on the geometry of the curve afteran appropriate embedding. Using algebraic-geometric methods,we then prove the conjecture through this approach in the caseof elliptic curves. In the process, we prove a new result aboutthe maximum number of points in an arc which lies on an ellipticcurve.     

Construction of MDS Self-dual Codes over Finite Fields

In this paper, we obtain some new results on the existence of MDS self-dual codes utilizing (extended) generalized Reed-Solomon codes over finite fields of odd characteristic. For finite field with odd characteristic and square cardinality, our results can produce more classes of MDS self-dual codes than previous works.     

Error-Correcting Codes over an Alphabet of Four Elements

The problem of finding the values of A_q(n,d)—the maximum size of a code of length n and minimum distance d over an alphabet of q elements—is considered. Upper and lower bounds on A₄(n,d) are presented and some values of this function are settled. A table of best known bounds on A₄(n,d) is given for n 12. When q M < 2q, all parameters for which A_q(n,d) = M are determined.     

Hyperplane Sections of Grassmannians and the Number of MDS Linear Codes

We obtain some effective lower and upper bounds for the number of (n,k)-MDS linear codes over _q. As a consequence, one obtains an asymptotic formula for this number. These results also apply for the number of inequivalent representations over _q of the uniform matroid or, alternately, the number of _q-rational points of certain open strata of Grassmannians. The techniques used in the determination of bounds for the number of MDS codes are applied to deduce several geometric properties of certain sections of Grassmannians by coordinate hyperplanes.     

The Structure of 1-Generator Quasi-Twisted Codes and New Linear Codes

One of the most important problems of coding theory is to construct codes with best possible minimum distances. Recently, quasi-cyclic (QC) codes have been proven to contain many such codes. In this paper, we consider quasi-twisted (QT) codes, which are generalizations of QC codes, and their structural properties and obtain new codes which improve minimum distances of best known linear codes over the finite fields GF(3) and GF(5). Moreover, we give a BCH-type bound on minimum distance for QT codes and give a sufficient condition for a QT code to be equivalent to a QC code.     

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.     

On the constructions of MDS self-dual codes via cyclotomy

MDS self-dual codes over finite fields have attracted a lot of attention in recent years by their theoretical interests in coding theory and applications in cryptography and combinatorics. In this paper we present a series of MDS self-dual codes with new length by using generalized Reed-Solomon codes and extended generalized Reed-Solomon codes as the candidates of MDS codes and taking their evaluation sets as a union of cyclotomic classes. The conditions on such MDS codes being self-dual are expressed in terms of cyclotomic numbers.     

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.     

Quasideterminant Characterization of MDS Group Codes over Abelian Groups

A group code defined over a group G is a subset of Gⁿ which forms a group under componentwise group operation. The well known matrix characterization of MDS (Maximum Distance Separable) linear codes over finite fields is generalized to MDS group codes over abelian groups, using the notion of quasideterminants defined for matrices over non-commutative rings.     

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.     

MDS and AMDS symbol-pair codes constructed from repeated-root cyclic codes

Symbol-pair codes introduced by Cassuto and Blaum in 2010 are designed to protect against the pair errors in symbol-pair read channels. One of the central themes in symbol-error correction is the construction of maximal distance separable (MDS) symbol-pair codes that possess the largest possible pair-error correcting performance. Based on repeated-root cyclic codes, we construct two classes of MDS symbol-pair codes for more general generator polynomials and also give a new class of almost MDS (AMDS) symbol-pair codes with the length lp. In addition, we derive all MDS and AMDS symbol-pair codes with length 3p, when the degree of the generator polynomials is no more than 10. The main results are obtained by determining the solutions of certain equations over finite fields.     

Symbol-Pair Distances of Repeated-Root Negacyclic Codes of Length 2s over Galois Rings

Negacyclic codes of length 2s over the Galois ring GR(2a,m) are linearly ordered under set-theoretic inclusion,i.e.,they are the ideals <(x + 1)i>,0 ≤ i ≤ 2sa,of the chain ring GR(2a,m)[x]/.This structure is used to obtain the symbol-pair distances of all such negacyclic codes.Among others,for the special case when the alphabet is the finite field F2m (i.e.,a =1),the symbol-pair distance distribution of constacyclic codes over F2m verifies the Singleton bound for such symbol-pair codes,and provides all maximum distance separable symbol-pair constacyclic codes of length 2s over F2m.     

Quantum MDS codes with new length and large minimum distance

According to the well-known CSS construction, constructing quantum MDS codes are extensively investigated via Hermitian self-orthogonal generalized Reed-Solomon (GRS) codes. In this paper, given two Hermitian self-orthogonal GRS codes $GR{S}_{k_1}(A,{\mathit{v}}_A)$ and $GR{S}_{k_2}(B,{\mathit{v}}_B)$, we propose a sufficient condition to ensure that $GR{S}_k(A\cup B,{\mathit{v}}_{A\cup B})$ is still a Hermitian self-orthogonal code. Consequently, we first present a new general construction of infinitely families of quantum MDS codes from known ones. Moreover, applying the trace function and norm function over finite fields, we give another two new constructions of quantum MDS codes with flexible parameters. It turns out that the forms of the lengths of our quantum MDS codes are quite different from previous known results in the literature. Meanwhile, the minimum distances of all the q-ary quantum MDS codes are bigger than $q/2+1$.