广义Hamming重量和码的等价

证明两个线性码之间的任一保持广义Hamming重量的线性同构是单项等价的, 推广了MacWilliams的一个重要定理.     

广义Hamming重量与码的半线性等价

设σ是域F/b的一个自同构,该文证明两个线性码之间的任一保持一个 广义Hamming重量的σ-半线性同构是一个σ-半线性单项等价.     

退化Hermite簇上射影码的广义Hamming重量

得到了有限域上射影空间中退化Hermite簇上射影码的广义Hamming重量,并计算了它们的广义重量谱.     

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。     

某些Hermitian码的广义Hamming权

本文对Ｈｅｒｍｉｔｉａｎ曲线上某些线性系统进行了仔细分析,从而给出了某些Ｈｅｒｍｉｔｉａｎ码第２及３个广义Ｈａｍｍｉｎｇ权的精确结果     

一些三重线性码的完备重量计数器

对于奇素数p,本文通过定义集的方法构造一些p元三重线性码,并利用有限域F_p上的Weil和,确定这些码的完备重量计数器.此外,证明这些码在某些条件下为极小码,进而适用于秘钥共享方案.特别地,得到一类参数为[p²-1, 3, p²-p-1]且达到Griesmer界的最优码.本文完善了简高鹏等在文献[1]中得到的部分结果.     

当$4q^{5}-5q^{4}-2q+1\leq d \leq 4q^{5}-5q^{4}-q$时$[g_{q}(6,d),6,d]_{q}$码的不存在性

证明了对于q≥17,当4q~5-5q~4-2q+1≤d≤4q~5-5q~4-q时,不存在达到Griesmer界的[n,k,d]_q码.此结果推广了Cheon等人在2005年和2008年的非存在性定理.     

最小距离固定且优于Gilbert—Varshamov界的码

本文构造了一类GF(q)上的码,其中GF(q)为q个元素的有限域.这些码的冗余取到渐进界r(q,n,7) 4 m,此界优于Gilbert-Varshamov存在界r(q,n,7) 5m.     

卡氏积码的r-MDR码与P_r-MDS码

本文研究了卡氏积码的r-广义Hamming重量计算公式和广义Singleton界,利用r-卡氏积码的子码仍为卡氏积码,证明了r-MDR码或Pr-MDR码的卡氏积码仍为r-MDR码或Pr-MDR码.同时也给出了这一个结果的部分逆命题.     

仿射簇上的码的参数 *

给出了仿射簇上的码的广义Hamming重量的一个下界 ,这类码由簇上适当的有理多项式列所定义 .     

On Generalized Hamming Weights for Galois Ring Linear Codes

The definition of generalized Hamming weights (GHW) for linear codes over Galois rings is discussed. The properties of GHW for Galois ring linear codes are stated. Upper and existence bounds for GHW of – linear codes and a lower bound for GHW of the Kerdock code over – are derived. GHW of some – linear codes are determined.     

Improvements on Generalized Hamming Weights of Some Trace Codes

We obtain improved bounds for the generalized Hamming weights of some trace codes which include a large class of cyclic codes over any finite field. In particular, we improve the corresponding bounds of Stichtenoth and Voss [8] using various methods altogether.     

Higher Weights of Anticodes and the Generalized Griesmer Bound

We show that the minimum r-weight d_r of an anticode can be expressed in terms of the maximum r-weight of the corresponding code. As examples, we consider anticodes from homogeneous hypersurfaces (quadrics and Hermitian varieties). In a number of cases, all differences (except for one) of the weight hierarchy of such an anticode meet an analog of the generalized Griesmer bound.     

On the Achievement of the Griesmer Bound

The main theorem in this paper is that there does not exist an [n,k,d]_q code with d = (k-2)q ^k-1 - (k-1)q^k-2 attaining the Griesmer bound for q k, k=3,4,5 and for q 2k-3, k 6.     

Minihypers and Linear Codes Meeting the Griesmer Bound: Improvements to Results of Hamada, Helleseth and Maekawa

This article improves results of Hamada, Helleseth and Maekawa on minihypers in projective spaces and linear codes meeting the Griesmer bound.In [10,12],it was shown that any -minihyper, with , where , is the disjoint union of points, lines,..., -dimensional subspaces. For q large, we improve on this result by increasing the upper bound on non-square, to non-square, square, , and (4) for square, p prime, p<3, to . In the case q non-square, the conclusion is the same as written above; the minihyper is the disjoint union of subspaces. When q is square however, the minihyper is either the disjoint union of subspaces, or the disjoint union of subspaces and one subgeometry . For the coding-theoretical problem, our results classify the corresponding codes meeting the Griesmer bound.     

第二类窃密信道中的组合数学方法

当信息在第二类窃密信道中传输时,线性码的广义汉明重量谱完全描述的它在该信道中的密码学特征.计算一个线性码的广义汉明重量谱是一个基本问题,首先提出了线性码的“最简基”的概念.在此基础上给出了一般线性码子码的几种计数公式,并给出了它们之间等价性的证明.     

Generalized Hamming weights of three classes of linear codes

The generalized Hamming weights of a linear code have been extensively studied since Wei first use them to characterize the cryptography performance of a linear code over the wire-tap channel of type II. In this paper, we investigate the generalized Hamming weights of three classes of linear codes constructed through defining sets and determine them partly for some cases. Particularly, in the semiprimitive case we solve a problem left in Yang et al. (2015) [30].     

Trellis Structure and Higher Weights of Extremal Self-Dual Codes

A method for demonstrating and enumerating uniformly efficient (permutation-optimal) trellis decoders for self-dual codes of high minimum distance is developed. Such decoders and corresponding permutations are known for relatively few codes.The task of finding such permutations is shown to be substantially simplifiable in the case of self-dual codes in general, and for self-dual codes of sufficiently high minimum distance it is shown that it is frequently possible to deduce the existence of these permutations directly from the parameters of the code.A new and tighter link between generalized Hamming weights and trellis representations is demonstrated: for some self-dual codes, knowledge of one of the generalized Hamming weights is sufficient to determine the entire optimal state complexity profile.These results are used to characterize the permutation-optimal trellises and generalized Hamming weights for all [32,16,8] binary self-dual codes and for several other codes. The numbers of uniformly efficient permutations for several codes, including the [24,12,8] Golay code and both [24,12,9] ternary self-dual codes, are found.