基于自对偶码的LDPC码构造（英文）

给出了一种通过自对偶码构造LDPC码的新方法.分析了此类码的码长及码率的范围.并通过MATLAB编程搜索找到了由在GF(11)上码长为16的自对偶码生成的一个具有较高码率和较好性质的LDPC码的实例.     

基于射影几何上的局部有向强正则图和LDPC码（英文）

有限域上的射影几何是有限几何中的一大类,其结构已被广泛研究.本文利用射影空间中的面构造了一个关联结构,给出了其为1(1/2)-设计的一个充要条件.另外,利用此关联结构得到了一个局部有向强正则图和LDPC码,并确定了它们的一些参数.     

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.     

有限域上一类常循环BCH码

本文研究了有限域F_q~2上一类码长为q^2m-1/r(q-1)的常循环BCH码,其中r|(q+1),q是素数幂.首先,给出了该类常循环BCH码是埃尔米特对偶包含码的一个充要条件.其次,确定这类埃尔米特对偶包含常循环BCH码的参数.最后,利用埃尔米特构造,得到了一些参数较好的量子码.     

有限环上r-MDR码与r-MDS码

本文研究了有限环上r-MDR码与r-MDS码.利用主理想环CRT（R₁,R₂,…,R_s）上的r-MDR码或P_r-MDS码CRT（C₁,C₂,…,C_s）,得到了某个链环R_i上的码C_i也是r-MDR码或P_r-MDR码.特别地,对于有限链环上的码C,给出了它的挠码Tor_i（C）为r-MDR码与r-MDS码的条件.     

环F2m+uF2m+u2F2m+u3F2m上线性码及其对偶码的Gray像

本文研究了环F2m+uF2m+u2 Fm+u3F2m上线性码.利用环是Frobenius环,证明了环上线性码C及其自对偶码的Gray像为F2m上的线性码和自对偶码.同时,给出了上循环码C的Gray像ψ(C)为F2m上的拟循环码.     

(z,u)-二元叠加码的卡氏积

(z,u)-码是编码理论中一种很重要的叠加码.利用两个已知的(z,u)-二元叠加码定义了它们的卡氏积,并计算了这个新(z,u)-二元叠加码的参数.     

Z4上偶长度的LCD负循环码

线性互补对偶(LCD)码是一类重要的纠错码,在通信系统、数据存储以及密码等领域都有重要的应用.文章研究了整数模4的剩余类环Z₄上偶长度的LCD负循环码,给出了这类码的生成多项式,证明了这类码是自由可逆码;并且利用Z₄上偶长度负循环码构造了一类Lee距离至少为6的LCD码.     

环R=F_4+vF_4上线性码的Macwilliams恒等式

本文研究了环R=F4+v F4上线性码及重量分布.利用环R=F4+v F4到F2的一种Gray映射?,证明了环上R线性码C的Gray像?(C)的对偶码为?(C⊥).然后,利用域F2上线性码与对偶码的重量分布的关系及Gray映射性质,给出了该环上线性码与对偶码之间的各种重量分布的Macwilliams恒等式.     

Z_4上偶长度的LCD负循环码

线性互补对偶(LCD)码是一类重要的纠错码,在通信系统、数据存储以及密码等领域都有重要的应用.文章研究了整数模4的剩余类环Z_4上偶长度的LCD负循环码,给出了这类码的生成多项式,证明了这类码是自由可逆码;并且利用Z_4上偶长度负循环码构造了一类Lee距离至少为6的LCD码.     

Linear codes with few weights from weakly regular plateaued functions

Linear codes with few nonzero weights have wide applications in secret sharing, authentication codes, association schemes and strongly regular graphs. Recently, Wu et al. (2020) obtained some few-weighted linear codes by employing bent functions. In this paper, inspired by Wu et al. and some pioneers' ideas, we use a kind of functions, namely, general weakly regular plateaued functions, to define the defining sets of linear codes. Then, by utilizing some cyclotomic techniques, we construct some linear codes with few weights and obtain their weight distributions. Notably, some of the obtained codes are almost optimal with respect to the Griesmer bound. Finally, we observe that our newly constructed codes are minimal for almost all cases.     

关于环$Z_4[u]/\langle u2-2\rangle$上的斜多元循环码

本文探索了环$R=Z_4[u]/\langle u2-2\rangle$ 上的几类斜多元循环码和多元循环码. 首先得到了环$R$上$(1,2u)$-多元循环码的生成多项式. 其次由定义的Gray映射得到了环$R$上$(1,2u)$- 多元循环码的Gray像是$Z_4$上的循环码或指数为2的逆循环码. 最后, 通过环$R$上$(1,2u)$- 多元循环码的一些例子来展示本文的主要结果.     

A bound on the minimum distance of generalized quasi-twisted codes

Generalized quasi-twisted (GQT) codes form a generalization of quasi-twisted (QT) codes and generalized quasi-cyclic (GQC) codes. By the Chinese remainder theorem, the GQT codes can be decomposed into a direct sum of some linear codes over Galois extension fields, which leads to the trace representation of the GQT codes. Using this trace representation, we first prove the minimum distance bound for GQT codes with two constituents. Then we generalize the result to GQT codes with s constituents. Finally, we present some examples to show that the bound is better than the well-known Esmaeili-Yari bound and sharp in many instances.     

On the Cyclicity of Goppa Codes,Parity-Check Subcodes of Goppa Codes,and Extended Goppa Codes

Classical Goppa codes are a special case of Alternant codes. First we prove that the parity-check subcodes of Goppa codes and the extended Goppa codes are both Alternant codes. Before this paper, all known cyclic Goppa codes were some particular BCH codes. Many families of Goppa codes with a cyclic extension have been found. All these cyclic codes are in fact Alternant codes associated to a cyclic Generalized Reed–Solomon code. In (1989, J. Combin. Theory Ser. A 51, 205–220) H. Stichtenoth determined all cyclic extended Goppa codes with this property. In a recent paper (T. P. Berger, 1999, in “Finite Fields: Theory, Applications and Algorithms (R. Mullin and G. Mullen, Eds.), pp. 143–154, Amer. Math. Soc., Providence), we used some semi-linear transformations on GRS codes to construct cyclic Alternant codes that are not associated to cyclic GRS codes. In this paper, we use these results to construct cyclic Goppa codes that are not BCH codes, new families of Goppa codes with a cyclic extension, and some families of non-cyclic Goppa codes with a cyclic parity-check subcode.     

Algebraic geometric codes with applications

The theory of linear error-correcting codes from algebraic geometric curves (algebraic geometric (AG) codes or geometric Goppa codes) has been well-developed since the work of Goppa and Tsfasman, Vladut, and Zink in 1981–1982. In this paper we introduce to readers some recent progress in algebraic geometric codes and their applications in quantum error-correcting codes, secure multi-party computation and the construction of good binary codes.      

Some New Results on Optimal Codes Over F5

We present some results on almost maximum distance separable (AMDS) codes and Griesmer codes of dimension 4 over over the field of order 5. We prove that no AMDS code of length 13 and minimum distance 5 exists, and we give a classification of some AMDS codes. Moreover, we classify the projective strongly optimal Griesmer codes over F₅ of dimension 4 for some values of the minimum distance.     

Construction of binary LCD codes,ternary LCD codes and quaternary Hermitian LCD codes

We give two methods for constructing many linear complementary dual (LCD for short) codes from a given LCD code, by modifying some known methods for constructing self-dual codes. Using the methods, we construct binary LCD codes and quaternary Hermitian LCD codes, which improve the previously known lower bounds on the largest minimum weights.

     

Systematic maximum sum rank codes

In the last decade there has been a great interest in extending results for codes equipped with the Hamming metric to analogous results for codes endowed with the rank metric. This work follows this thread of research and studies the characterization of systematic generator matrices (encoders) of codes with maximum rank distance. In the context of Hamming distance these codes are the so-called Maximum Distance Separable (MDS) codes and systematic encoders have been fully investigated. In this paper we investigate the algebraic properties and representation of encoders in systematic form of Maximum Rank Distance (MRD) codes and Maximum Sum Rank Distance (MSRD) codes. We address both block codes and convolutional codes separately and present necessary and sufficient conditions for an encoder in systematic form to generate a code with maximum (sum) rank distance. These characterizations are given in terms of certain matrices that must be superregular in a extension field and that preserve superregularity after some transformations performed over the base field. We conclude the work presenting some examples of Maximum Sum Rank convolutional codes over small fields. For the given parameters the examples obtained are over smaller fields than the examples obtained by other authors.     

A class of almost MDS codes

MDS codes and almost MDS (AMDS) codes are special classes of linear codes, and have important applications in communications, data storage, combinatorial theory, and secrete sharing. The objective of this paper is to present a class of AMDS codes from some BCH codes and determine their parameters. It turns out the proposed AMDS codes are distance-optimal and dimension-optimal locally repairable codes. The parameters of the duals of this class of AMDS codes are also discussed.