二元等距码的Q-变换分布及其应用

研究了二元等距码、等重等距码及其距离分布的Q-变换。通过使用Q-变换分布的性质,研究了二元等距码和等重等距码的最大码字数并得到2个新的上界,这些上界在某些情况下优于已知的结果。     

关于等重码不可检错误概率的界

本文研究了二元等重码不可检错误概率(UEP)的界.首先,我们通过研究二元等重码的对偶距离分布及其性质,给出二元等重码UEP的一个新的下界,该下界改进了Fu-Kl  ve-Wei的最新结果;然后,我们指出2003年Fu-Kl  ve-Wei关于二元等重码UEP上界的某些结果有错误,我们随后给出更正后的结果,即二元等重码UEP的平均值和一个上界.     

等重码的一些新结果

本文给出两种构造二元非线性等重码的方法，这些方法是［１］文构造二元非线性循环等重码方法的改进。通过我们的构造方法可以得到几类二元最优等重码。我们进一步说明通过ＧＦ（ｑ）上的等重码和达到Ｐｌｏｔｋｉｎ界的最优码可以构造达到Ｊｏｈｎｓｏｎ上界的二元最优等重码     

再论二元非线性等重码的检错性能

本文利用对偶距离分布的性质,从理论上证明了文「４」的必要条件在δ＝１时实际上是充分条件,并得出对于（ｎ,２,ω）码,最佳检错码与检错好码是等价的,最后,本文讨论了（ｎ,２,ω）码不可检错误概率的界。     

线性拟等重码的结构分析

Ｃ为二元正侧线性拟等重码，我们证明：（１）如果ｎ＝２ｄ则Ｃ等价于１阶；（２）如何为素数，则Ｃ等价于ＲＭ（ｋ－１）删除第１个分量后得到的线性码，另外，利用编码理论中著名的Ｍａｃ Ｗｉｌｌｉａｍ恒等式给出文（１）定量１的一个新的简洁证明。     

关于等重码的对偶距离

本文借助于对偶距离和相关免疫阶的关系，证明了等重码的对偶距离为１或２。     

广义Hamming线性等重码

设C为 线性码,本文证明:一个线性码C只要对于某个r,0＜r＜k,是r-等重线性码,那么它对于所有的0＜r＜k都是r-等重线性码.     

一类达到上界的拉丁方和正交拉丁方等重码

证明了一类能够达到Johson上界的特殊拉丁方和正交拉丁方所对应的等重码的存在性。讨论了其存在条件,导出了有关基本公式,并给出实例设计和该码的检错性能分析。     

非线性等距码和等距等重码的上界

设Qq(n,d)代表码长为n、任意两个不同码字间的Hamming距离为d的q元等距码所能达到的最大可能码字数(不考虑码的重量);Eq(n,d,w)代表码长为n、任意两个不同码字间Ham-ming距离为d、每个码字重量为w的q元等距等重码所能达到的最大可能码字数量.设q,n,d,w∈N,获得当q＞2时,有①Eq(n,d,w)≤qn,②Qq(n,d)≤qn+1;当q=2时,则有③Eq(n,d,w)≤n,④Qq(n,d)≤n+1.     

广义Hamming重量和等重码

本文将线性码的广义Hamming重量的概念推广到非线性码上去,并导出了一种广义Elias界.对于线性等重码,本文给出了其完整的重量谱系.     

Heuristic Algorithms for Constructing Binary Constant Weight Codes

Constant weight binary codes are used in a number of applications. Constructions based on mathematical structure are known for many codes. However, heuristic constructions unrelated to any mathematical structure can become of greater importance when the parameters of the code are larger. This paper considers the problem of finding constant weight codes with the maximum number of codewords from a purely algorithmic perspective. A set of heuristic and metaheuristic methods is presented and developed into a variable neighborhood search framework. The proposed method is applied to 383 previously studied cases with lengths between 29 and 63. For these cases it generates 153 new codes, with significantly increased numbers of codewords in comparison with existing constructions. For 10 of these new codes the number of codewords meets a known upper bound, and so these 10 codes are optimal. As well as the ability to generate new best codes, the approach has the advantage that it is a single method capable of addressing many sets of parameters in a uniform way.     

On the Number of Inequivalent Binary Self-Orthogonal Codes

Let Psi_k,n denote the number of inequivalent binary self-orthogonal [n, k] codes. We present a method which allows us to compute Psi_k,n explicitly for a moderate k and an arbitrary n. Included in this paper are explicit formulas for Psi_k,n with k les 5.     

最佳非线性等重检错码的存在性问题

本文分析了二进制（２ｍ，２，ｍ）非线性等重码的不可检错误概率，证明了当ｍ≥５时，它不是最佳检错码，从而完全证明了作者提出的：当ｍ≥９时，不存在最佳二进制非线性等重码的猜想。     

Determination of the Local Weight Distribution of Binary Linear Block Codes

Some methods to determine the local weight distribution of binary linear codes are presented. Two approaches are studied: A computational approach and a theoretical approach. For the computational approach, an algorithm for computing the local weight distribution of codes using the automorphism group of the codes is devised. In this algorithm, a code is considered the set of cosets of a subcode, and the set of cosets is partitioned into equivalence classes. Thus, only the weight distributions of zero neighbors for each representative coset of equivalence classes are computed. For the theoretical approach, relations between the local weight distribution of a code, its extended code, and its even weight subcode are studied. As a result, the local weight distributions of some of the extended primitive Bose-Chaudhuri-Hocquenghen (BCH) codes, Reed-Muller codes, primitive BCH codes, punctured Reed-Muller codes, and even weight subcodes of primitive BCH codes and punctured Reed-Muller codes are determined     

一种构造GC常重量DNA码的方法

GC重量是DNA码的一个重要参数,如何构造满足GC常重量约束的DNA码是一个有趣的问题。该文通过在DNA码与四元码之间建立一个双射,将构造满足GC常重量约束的DNA码转化为构造GC常重量四元码。通过代数的方法,构造了3类满足GC常重量约束的DNA码。