On the Weight Hierarchy of Product Codes

Let V and W be codes and let C = V W be the product code of V and W. In [6] Wei and Yang, conjectured a formula for the generalized Hamming weights of V W in terms of those of V and W provided that both V and W satisfy the chain condition. Recently the conjecture has been proved by Schaathun [4]. In this paper we generalize the formula to a product code with more than two components.     

3维11元线性码的重量谱

Chen,Kl(o)ve利用有限射影几何的方法,提出了四个有效的结构,确定了3维q元线性码的几乎所有重量谱.本文将结构4应用于不满足链条件的3维11元线性码,还剩13类未知序列;并运用有限射影几何的方法,提出了一种新的交点赋零结构,最终确定了这13类未知序列对应的重量谱,从而得到了不满足链条件的3维11元线性码全部可能的重量谱.     

4维3元断链码的重量谱

GF(q)上[n,k;q]线性码C的重量谱为序列(d1,d2,…,dk),这里dr是C的r维子码的最小支持重量(1≤r≤k).用有限射影几何方法确定了满足含有2个邻接断点的断链条件的4维3元线性码的重量谱.     

Finite Projective Geometries and Classification of the Weight Hierarchies of Codes （Ⅰ）

The weight hierarchy of a binary linear [n,κ] code C is the sequence (d1,d2,...,dκ), where dr is the smallest support of an r-dimensional subcode of C. The codes of dimension 4 are collected in classes and the possible weight hierarchies in each class is determined by finite projective geometries.The possible weight hierarchies in class A, B, C, D are determined in Part (Ⅰ). The possible weight hierarchies in class E, F, G, H, I are determined in Part (Ⅱ).     

一类4维3元线性码的重量谱

GF(q)上[n,k;q]线性码C的重量谱为序列(d1,d2,…dk),其中dr是C的r维子码的最小支持重量.文章利用有限射影几何方法确定了一类4维3元线性码的重量谱,并对其进行了验证.     

关于外积码的Hamming谱

本文研究了任意有限域Fq上的两个线性码的外积及其有关性质；并给出了由两个线性码构造的外积码的Hamming谱的第1个谱值的界以及最后一个谱值。     

On Perfect Ternary Constant Weight Codes

We consider the space of ternary words of length n and fixed weight w with the usual Hamming distance. A sequence of perfect single error correcting codes in this space is constructed. We prove the nonexistence of such codes with other parameters than those of the sequence.     

满足右零化子特殊升链条件的环

本文讨论了满足右零化子特殊升链条件的环的性质以及与左、右完备环，ＱＦ－环的关系；给出了满足右零化子特殊升链条件的环中素理想是完全素理想的条件，从而给出了Ｇｏｌｄｉｅ定理的一个应用．     

Linear Codes and Their Coordinate Ordering

A necessary and sufficientcondition for a q-ary code to satisfy the two-waychain condition (TCC) is found. A known construction of q-arycodes is shown to yield codes satisfying the TCC. Some q-arycodes of dimension k 6 meeting the Griesmerbound are proved to satisfy the TCC.     

The Weight Enumerator of Linear Codes over GF q m) Having Generator Matrix over GF q

We have the relationships between the Hamming weight enumerator of linear codes over GFq ^m which have generator matrices over GFq, the support weight enumerator and the -ply weight enumerator.     

On a Class of Traceability Codes

Traceability codes are designed to be used in schemes that protect copyrighted digital data against piracy. The main aim of this paper is to give an answer to a Staddon–Stinson–Wei's problem of the existence of traceability codes with q< w ² and b>q. We provide a large class of these codes constructed by using a new general construction method for q-ary codes.     

On the Linear Codes Arising from Schubert Varieties

We prove a lower bound for the minimum distance of the linear code associated to a Schubert variety. Moreover for a Schubert variety of lines we compute the full distribution of weights.     

一类小缺陷码的链条件

利用射影几何方法在小缺陷码中,NMDS码是链条件码;给出k维NμMDS(0μk-2)码满足链条件的一个充要条件与一些易判断的充分条件.     

Maximal Weight Divisors of Projective Reed-Muller Codes

Projective Reed-Muller (PRM) codes, as the name suggests, are the projective analogues of generalized Reed-Muller codes. The parameters are known, and small steps have been taken towards pinning down the codeword weights that occur in any PRM code. We determine, for any PRM code, the greatest common divisor of its codeword weights.     

Automorphisms of Constant Weight Codes and of Divisible Designs

We show that the automorphism group of a divisible design is isomorphic to a subgroup H of index 1 or 2 in the automorphism group of the associated constant weight code. Only in very special cases H is not the full automorphism group.     

Antichain Codes

We show that almost all codes satisfy an antichain condition. This states that the minimum length of a two dimensional subcode of a code C increases if the subcode is constrained to contain a minimum weight codeword. In particular, almost no code satisfies the chain condition. In passing, we study the typical behaviour of codes with respect to generalized distances and show that almost all lie on a generalized Varshamov-Gilbert bound.     

On the Minimum Weight of Codes over F5 Constructed from Certain Conference Matrices

In this paper, codes over F₅ with parameters [36, 18, 12], [48, 24, 15], [60, 30, 18], [64, 32, 18] and [76, 38, 21] which improve the previously known bounds on the minimum weight for linear codes over F₅ are constructed from conference matrices. Through shortening and truncating, the above codes give numerous new codes over F₅ which improve the previously known bounds on minimum weights.     

有限链环上的循环码及其Mattson-Solomn多项式

研究了有限链环上的循环码的结构及其Mattson-Solomn多项式,用循环码的Mattson-Solomn多项式和定义集刻画循环码及其对偶码的性质。     

A Lower Bound on the Greedy Weights of Product Codes

A greedy 1-subcode is a one-dimensional subcode of minimum (support) weight. A greedy r-subcode is an r-dimensional subcode with minimum support weight under the constraint that it contain a greedy (r - 1)-subcode. The r-th greedy weight e _r is the support weight of a greedy r-subcode. The greedy weights are related to the weight hierarchy. We use recent results on the weight hierarchy of product codes to develop a lower bound on the greedy weights of product codes.