The Sizes of Optimal q-Ary Codes of Weight Three and Distance Four: A Complete Solution
Yeow Meng Chee, Son Hoang Dau, Alan C. H. Ling, San Ling
这封信介绍了两个新的建设性的技术，以完成确定的最佳q元码的大小等重量3和距离4。
This correspondence introduces two new constructive techniques to complete the determination of the sizes of optimal q-ary codes of constant weight three and distance four.
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New completely regular q-ary codes based on Kronecker products
对任意整数$ρ\geq 1$和任意素数幂q，给出了一个覆盖半径为$ρ$，d=3的完全正则（且完全传递）q元码的无限族的显式构造.还计算交集数组。在同样的条件下，给出了覆盖半径为$ρ$的非完全正则的q元均匀填充码（广义）的显式构造.在这两种构造中，克罗内克积都是使用的基本工具。
Weight Distribution of A p-ary Cyclic Code
对于一个奇素数$p$和两个正整数$n\geq 3$和$k$，其中$\frac{n}{{\rm gcd}(n,k)}$为奇，本文确定了$p$元循环码$\mathcal{C}$在$\mathbb{F}_{p}$上的重量分布，其中$α$为$\mathbb{F}_{p^n}$的本原元，$α^{-1}$、$α^{-(p^k+1)}$和$α^{-(p^{3k}+1)}$均为非零
Constructions of q-Ary Constant-Weight Codes
本文提出了一种新的q元等重码的组合构造方法，它产生了几类最优码和渐近最优码。该构造揭示了q元等重码与各种类型的成对不相交组合设计之间的密切联系。
The weight distributions of two classes of p ary cyclic codes with few weights
循环码由于具有高效的编码和译码算法，几十年来吸引了大量的研究兴趣。本文对奇素数$p$，完全确定了两类$p$ -元循环码的重量分布。我们表明，这两个代码有最多五个非零的重量。
On linear $q$-ary completely regular codes with $ρ=2$ and dual antipodal
当对偶码对映时，刻画了所有覆盖半径为$ρ=2$的线性$q$ -ary完全正则码.这些完全正则码是覆盖半径为1的线性完全正则码的推广，它们都是可分类的。对于$ρ=2$，我们给出了我们已知的所有此类代码的列表。这也给出了两个重量线性对跖码的特征。
Gray Codes and Overlap Cycles for Restricted Weight Words
格雷码是一组组合对象的列表结构，使得在列表中的所有相邻元素中保持某种一致（通常是最小的）变化属性。虽然格雷码的m进制字符串已被认为是在过去，我们提供了一个新的，简单的格雷码固定重量的m进制字符串。此外，我们考虑了一种相对较新的类型的格雷码称为重叠周期，并证明了基本存在的结果，关于重叠周期的固定重量和重量范围的m元字。
The weight distribution and weight hierarchy of linear codes are two important research topics in coding theory. In this paper, by choosing proper defining sets from inhomogeneous quadratic functions over $\mathbb{F}_{q}^{2},$ we construct a family of $3$-weight $p$-ary linear codes and determine their weight distributions and weight hierarchies. Most of the codes can be used in secret sharing schemes.
Linear Size Optimal q-ary Constant-Weight Codes and Constant-Composition Codes
一个最优的等重码$w$具有线性长度当且仅当它的距离$d$至少为$2w-1$。当$d\geq 2w$时，确定这种恒定组成或恒定重量代码的确切大小是微不足道的，但是$d=2w-1$的情况先前仅针对二进制和三进制恒定组成和恒定重量代码以及针对一些零星的实例而被解决。基于差三角集的一种新的推广，给出了权$w$和距离$2w-1$的拟循环最优常合成常重码的构造.结果，对于所有这样的长度足够大的码，确定了权重$w$和距离$2w-1$的最佳恒定组成码和最佳恒定权重码的大小。这解决了Etzion的一个开放问题。除了两种情况外，还确定了所有$w\leq 6$的重量$w$和距离$2w-1$的最佳常数组合码的大小。
We present an encoding and decoding scheme for constant weight sequences, that is, given an information sequence, the construction results in a sequence of specific weight within a certain range. The scheme uses a prefix design that is based on Gray codes. Furthermore, by adding redundant symbols we extend the range of weight values for output sequences, which is useful for some applications.
Linear codes with a few weights are very important in coding theory and have attracted a lot of attention. In this paper, we present a construction of $q$-ary linear codes from trace and norm functions over finite fields. The weight distributions of the linear codes are determined in some cases based on Gauss sums. It is interesting that our construction can produce optimal or almost optimal codes. Furthermore, we show that our codes can be used to construct secret sharing schemes with interesting access structures and strongly regular graphs with new parameters.
Relative generalized Hamming weights of q-ary Reed-Muller codes
$q$元Reed-Muller码的陪集构造可以用于在分布式存储系统上存储秘密，使得只有访问系统的大部分的方可以获得信息，同时仍然允许本地纠错。在本文中，我们确定了这些码的相对广义汉明重量，可以转化为信息泄漏的详细描述[2，24，21，11]。
Pseudocodeword weights for non-binary LDPC codes
研究了q元LDPC码的伪码字，定义了伪码字在q元对称信道上的权值。AWGN信道上的伪码字的权重定义也被扩展到二维q元调制，如q-PAM和q-PSK。基于树的最小伪码字权重的下限也保持在这些信道上的q元LDPC码。
The weight distribution of a family of p-ary cyclic codes
设m，k是正整数，p是奇素数，$π$是$\mathbb{F}_{p^m}$的本原元.本文确定了$\mathbb{F}_p$上的循环码族$\mathcal{C}_t$的重量分布，其中$\mathbb{F}_p$上有两个零点$π^{-t}$和$-π^{-t}$，且$t$对某些$τ\in \{0,1,\cdots, m-1\}$满足$t\equiv \frac{p^k+1}{2}p^τ\ ({\rm mod}\ \frac{p^m-1}{2}) $.
The weight distribution of second order $q$-ary Reed-Muller codes have been determined by Sloane and Berlekamp (IEEE Trans. Inform. Theory, vol. IT-16, 1970) for $q=2$ and by McEliece (JPL Space Programs Summary, vol. 3, 1969) for general prime power $q$. Unfortunately, there were some mistakes in the computation of the latter one. This paper aims to provide a precise account for the weight distribution of second order $q$-ary Reed-Muller codes. In addition, the weight distributions of second order $q$-ary homogeneous Reed-Muller codes and second order $q$-ary projective Reed-Muller codes are …
Weight hierarchies of 3-weight linear codes from two $p$-ary quadratic functions
本研究以两个有限域上二次型构造的集合为定义集，构建一类3重量$p$元线性码，完全确定其重量分布与重量谱系，多数码可应用于秘密共享方案。
Weight hierarchies of three-weight $p$-ary linear codes from inhomogeneous quadratic forms
本文基于非齐次二次型构造了一族三权$p$元线性码，并确定了其重量分布与权层级，这些码大多适用于秘密共享方案。
Local-Testability and Self-Correctability of q-ary Sparse Linear Codes
我们证明了具有小偏差的q元稀疏码是自校正的和局部可测的。推广了Kaufman和Sudan的一个结果，证明了具有小偏差的二元稀疏码的局部可测性和可纠正性。我们使用q元Krawtchouk多项式和McWilliams恒等式的属性-将代码的权重分布与其对偶的权重分布联系起来-来推导我们正在分析的随机测试器和自校正器的错误概率的界限。
In this paper, several classes of three-weight codes and two-weight codes for the homogeneous metric over the chain ring $R=\mathbb{F}_p+u\mathbb{F}_p+\cdots +u^{k-1}\mathbb{F}_{p},$ with $u^k=0,$ are constructed, which generalises \cite{SL}, the special case of $p=k=2.$ These codes are defined as trace codes. In some cases of their defining sets, they are abelian. Their homogeneous weight distributions are computed by using exponential sums. In particular, in the two-weight case, we give some conditions of optimality of their Gray images by using the Griesmer bound. Their dual homogeneous dis…
Research on Linear Codes Holding $q$-Ary $t$-Designs
本文研究线性码是否能生成q-ary t-设计，提出两种新准则，并展示多种线性码生成q-ary 2-设计，同时提供基于码自动群的替代方法。
It was shown in \cite{GXY18} that the length $n$ of a $q$-ary linear locally recoverable code with distance $d\ge 5$ is upper bounded by $O(dq^3)$. Thus, it is a challenging problem to construct $q$-ary locally recoverable codes with distance $d\ge 5$ and length approaching the upper bound. The paper \cite{GXY18} also gave an algorithmic construction of $q$-ary locally recoverable codes with locality $r$ and length $n=Ω_r(q^2)$ for $d=5$ and $6$, where $Ω_r$ means that the implicit constant depends on locality $r$. In the present paper, we present an explicit construction of $q$-ary locally re…
Recently, linear codes with a few weights were widely investigated due to their applications in secret sharing schemes and authentication schemes. In this letter, we present a class of $q$-ary linear codes derived from irreducible cyclic codes with $q$ a prime power. We use Gauss sums to represent its Hamming weights and obtain the bounds of its minimum Hamming distance. In some cases, we explicitly determine its weight distributions which generalize some known results. It is quite interesting that many new codes obtained in this letter are optimal according to some bounds on linear codes.
In this paper, several classes of three-weight codes and two-weight codes for the homogeneous metric over the chain ring $R=\mathbb{F}_p+u\mathbb{F}_p+\cdots +u^{k-1}\mathbb{F}_{p},$ with $u^k=0,$ are constructed, which generalises \cite{SL}, the special case of $p=k=2.$ These codes are defined as trace codes. In some cases of their defining sets, they are abelian. Their homogeneous weight distributions are computed by using exponential sums. In particular, in the two-weight case, we give some conditions of optimality of their Gray images by using the Griesmer bound. Their dual homogeneous dis…
Weight enumeration of codes from finite spaces
本文研究了q元Simplex码和q元一阶Reed-Muller码的广义和扩展的重量计数器。对于我们的计算，我们使用这些代码对应于一个投影系统，包含在一个有限的投影或仿射空间的所有点。由于我们使用的几何方法的重量枚举，我们也完全确定了一个扩展码的子码和字的支持集。
Sharper upper bounds for $q$-ary and constant-weight $B_2$ codes
本文通过利用差分布的傅里叶结构，推导出更紧的 $q$-ary $B_2$ 代码熵上界，并结合前缀-后缀方法改进了渐进行率上界。同时研究了二元常权 $B_2$ 代码的上界。
Fastest Distributed Consensus Averaging Problem on Perfect and Complete n-ary Tree networks
解决最快的分布式共识平均问题（即，在具有不同拓扑结构的网络上找到边上的权重以最小化权重矩阵的第二大特征值模）是传感器网络领域中的主要研究领域之一，并且在这个问题中众所周知的网络之一是树形网络。在这项工作中，我们提出了最快的分布式共识平均算法的分层和半定规划的方法，为两种特殊类型的树网络，即完美和完整的n元树网络的问题的解析解。该方法基于最快分布式共识平均问题的凸性，通过对松弛条件下的特征多项式进行归纳比较，找到最优权值。此外，某些类型的分支（例如完美和完整的n元树分支）的边的最佳权重独立于网络的其余部分来确定。
The paper deals with the perfect 1-error correcting codes over a finite field with $q$ elements (briefly $q$-ary 1-perfect codes). We show that the orthogonal code to the $q$-ary non-full-rank 1-perfect code of length $n = (q^{m}-1)/(q-1)$ is a $q$-ary constant-weight code with Hamming weight equals to $q^{m - 1}$ where $m$ is any natural number not less than two. We derive necessary and sufficient conditions for $q$-ary 1-perfect codes of non-full rank. We suggest a generalization of the concatenation construction to the $q$-ary case and construct the ternary 1-perfect codes of length 13 and …
Linear codes with a few weights are an important class of codes in coding theory and have attracted a lot of attention. In this paper, we present several constructions of $q$-ary linear codes with two or three weights from vectorial dual-bent functions, where $q$ is a power of an odd prime $p$. The weight distributions of the constructed $q$-ary linear codes are completely determined. We illustrate that some known constructions in the literature can be obtained by our constructions. In some special cases, our constructed linear codes can meet the Griesmer bound. Furthermore, based on the const…
A class of $p$-ary cyclic codes and their weight enumerators
设$m$，$k$是正整数，使得$\frac{m}{\gcd(m,k)}\geq 3$，$p$是奇素数，$π$是$\mathbb{F}_{p^m}$的本原元.令$h_1(x)$和$h_2(x)$分别是$-π^{-1}$和$π^{-\frac{p^k+1}{2}}$在$\mathbb{F}_p$上的最小多项式。在$\frac{m}{\gcd(m,k)}$为奇数、$k$为偶数、$\gcd(m,k)$为奇数或$\frac{k}{\gcd(m,k)}$为奇数的情况下，Zhou等人在\cite{zhou}中得到了一类循环码$\mathcal{C}$在$\mathbb{F}_p$上的具有奇偶校验多项式$h_1(x)h_2(x)$的重量分布.本文进一步研究了$\mathbb{F}_p$上的这类循环码$\mathcal{C}$在其余情况为奇数$\frac{m}{\gcd(m,k)}$和偶数情况为$\frac{m}{\gcd(m,k)}$的情形.此外，我们确定了循环码$\mathcal{C}$的重量分布。
A Further Investigation on Complete Complementary Codes from $q$-ary Functions
本研究首次确立了$q$元函数的充要条件以构造具有灵活参数的完全互补码，并证明$ν$元函数在特定互质条件下能生成集合大小和字母表大小更灵活的CCCs。
Berlekamp and Zigangirov completely determined the capacity error function for binary error correcting codes with noiseless feedback. It is still an unsolved problem if the upper bound for the capacity error function in the non-binary case of Ahlswede, Lebedev, and Deppe is sharp. We consider wraparound channels with limited magnitude and noiseless feedback. We completely determine the capacity error function for all $q$-ary wraparound channels with a magnitude of level $r$. All of our algorithms use partial noiseless feedback. Furthermore, a special case of the problem is equivalent to Shanno…
Codes that can correct up to $t$ symmetric errors and detect all unidirectional errors, known as $t$-EC-AUED codes, are studied in this paper. Given positive integers $q$, $a$ and $t$, let $n_q(a,t+1)$ denote the length of the shortest $q$-ary $t$-EC-AUED code of size $a$. We introduce combinatorial constructions for $q$-ary $t$-EC-AUED codes via one-factorizations of complete graphs, and concatenation of MDS codes and codes from resolvable set systems. Consequently, we determine the exact values of $n_q(a,t+1)$ for several new infinite families of $q,a$ and $t$.
A Butson-Hadamard matrix H is a square matrix of dimension n whose entries are complex roots of unity such that HH*= nI. In the first part of this work, some new results on generalized Gray map are studied. In the second part, codes obtained from Butson-Hadamard matrices and some bounds on the minimum distance of these codes are proved. In particular, we show that the code obtained from a Butson-Hadamard matrix meets the Plotkin bound under a non-homogeneous weight. We also give the parameters of some code families which are obtained from modified Butson-Hadamard matrices under a (non)homogene…
Further Results on the Classification of MDS Codes
在大小为$q$的字母表$\mathcal{A}$上具有长度$n$、维度$k$的$q$元最大距离可分离（MDS）码$C$是作为$\mathcal{A}^n$的元素的$q^k$码字的集合，使得$C$中的两个不同码字之间的汉明距离至少为$n-k+1$。在早期的研究中，已经对$q\leq 9$阶的相互正交拉丁方的集合进行了分类，其对应于二维\mbox{$q$-}元MDS码和用于$q\leq 8$的$q$元一错误校正MDS码。这些结果在这里被用来完成分类的所有$7$ -ary和$8$ -ary MDS代码与$d\geq 3$使用计算机搜索。
Upper and lower bounds on the largest number of weights in a cyclic code of given length, dimension and alphabet are given. An application to irreducible cyclic codes is considered. Sharper upper bounds are given for the special cyclic codes (called here strongly cyclic), {whose nonzero codewords have period equal to the length of the code}. Asymptotics are derived on the function $Γ(k,q),$ {that is defined as} the largest number of nonzero weights a cyclic code of dimension $k$ over $\F_q$ can have, and an algorithm to compute it is sketched. The nonzero weights in some infinite families of R…
Four classes of optimal p-ary cyclic codes
本文提出了四类最优p-ary循环码，其中三类为无限类，并展示了已知的五进制循环码是其特例。
Binary Multiple-Node-Erasure-Correcting Codes over Complete Graphs: Constructions, q-Ary Metric Balls, and Duality
该研究针对完全图上的二元多节点删除纠正码，扩展循环构造并推导相关度量球公式，得到特定参数下的最优码及对偶性结果。
New $q$-ary Quantum MDS Codes with Distances Bigger than $\frac{q}{2}$
量子MDS码的构造已经被许多作者研究过。我们参考[3]第1482页的表格了解已知的结构。然而，对于稀疏长度$n>q+1$，只有很少的$q$元量子MDS $[[n,n-2d+2,d]]_q$码具有最小距离$d>\frac{q}{2}$。在$n=\frac{q^2-1}{m}$的情况下，其中$m|q+1$或$m|q-1$有完整的结果。在其中$m|q^2-1$不是$q-1$或$q+1$的因子的$n=\frac{q^2-1}{m}$的情况下，没有构造具有$d> \frac{q}{2}$的$q$元量子MDS码。本文提出了一种直接构造${\bf F}_{q^2}$上Hermitian自正交码的方法。在这种情况下，我们给出了一些新的$q$ -ary量子码.此外，我们还提出了许多新的长度为$\frac{w(q^2-1)}{u}$和最小距离为$d > \frac{q}{2}$的$q$元量子MDS码.
Self-Orthogonal Codes from Vectorial Dual-Bent Functions
本文利用向量对偶弯曲函数构造新的q元自正交码族，确定部分重量分布，导出至少几乎最优的线性码、LCD码和量子码，并统一了已有构造。
We construct two new infinite families of trace codes of dimension $2m$, over the ring $\mathbb{F}_p+u\mathbb{F}_p,$ when $p$ is an odd prime. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using Gauss sums. By Gray mapping, we obtain two infinite families of linear $p$-ary codes of respective lengths $(p^m-1)^2$ and $2(p^m-1)^2.$ When $m$ is singly-even, the first family gives five-weight codes. When $m$ is odd, and $p\equiv 3 \pmod{4},$ the first family yields $p$-ary two-weight codes, which are shown to be optimal by application of the Gries…
Tree-Based Construction of LDPC Codes Having Good Pseudocodeword Weights
本文提出了一种基于树的LDPC码构造方法，该方法的最小伪码字权值等于或几乎等于最小距离，并具有良好的迭代译码性能。该构造涉及枚举固定层数的$d$ -正则树，并采用基于排列或相互正交的拉丁方的连接算法来关闭树。给出了度$d=p^s$和度$d = p^s+1$的方法，其中$p$是素数.一类对应于著名的有限几何和有限广义四边形LDPC码，提出的其他代码是新的。我们还提出了$p$ -ary LDPC码的伪码字重量的一些界限。将这些码作为$p$ -元LDPC码而不是二进制LDPC码来处理，提高了它们的速率、最小距离和伪码字权重，从而赋予有限几何LDPC码新的重要性，其中$p > 2$。
Hyperovals in $\PG(2,\gf(q))$ with even $q$ are maximal arcs and an interesting research topic in finite geometries and combinatorics. Hyperovals in $\PG(2,\gf(q))$ are equivalent to $[q+2,3,q]$ MDS codes over $\gf(q)$, called hyperoval codes, in the sense that one can be constructed from the other. Ovals in $\PG(2,\gf(q))$ for odd $q$ are equivalent to $[q+1,3,q-1]$ MDS codes over $\gf(q)$, which are called oval codes. In this paper, we investigate the binary subfield codes of two families of hyperoval codes and the $p$-ary subfield codes of the conic codes. The weight distributions of these …
Recently, linear codes with few weights have been constructed and extensively studied. In this paper, for an odd prime p, we determined the complete weight enumerator of two classes of p-ary linear codes constructed from defining set. Results show that the codes are at almost seven-weight linear codes and they may have applications in secret sharing schemes.
A Family of Five-Weight Cyclic Codes and Their Weight Enumerators
循环码是线性码的子类，并且由于它们具有有效的编码和解码算法而在消费电子、数据存储系统和通信系统中具有应用。本文提出了一类具有三个零点的$p$元循环码。确定了这类循环码的重量分布。事实证明，所提出的循环码有五个非零的权重。
The Weight Enumerator of Three Families of Cyclic Codes
循环码是线性码的子类，并且由于其高效的编码和解码算法而在消费电子、数据存储系统和通信系统中具有广泛的应用。具有多个零点的循环码及其对偶码是多年来的研究课题。然而，它们的权重分布仅在极少数情况下已知。一般情况下，计算循环码的重量分布是很大程度上基于有限域上的一些指数和的评估。最近，Li，Hu，Feng和Ge研究了一类长度为$p^{2m}-1$的$p$ -元循环码，其中$p$是素数，$m$是奇数.他们通过在所涉及的指数和与埃尔米特形式图的谱之间建立联系来确定这类循环码的重量分布。本文推广了这类$p$ -元循环码，并利用Li，Hu，Feng，Ge的思想，解决了这类广义循环码的重量分布问题，其中包括偶数$m$和奇数$m$.还确定了两个相关循环码族的重量分布。
Recently, linear codes constructed from defining sets have been investigated extensively and they have many applications. In this paper, for an odd prime $p$, we propose a class of $p$-ary linear codes by choosing a proper defining set. Their weight enumerators and complete weight enumerators are presented explicitly. The results show that they are linear codes with three weights and suitable for the constructions of authentication codes and secret sharing schemes.
Cyclic codes have attracted a lot of research interest for decades. In this paper, for an odd prime $p$, we propose a general strategy to compute the complete weight enumerator of cyclic codes via the value distribution of the corresponding exponential sums. As applications of this general strategy, we determine the complete weight enumerator of several $p$-ary cyclic codes and give some examples to illustrate our results.
Linear codes have been an interesting topic in both theory and practice for many years. In this paper, a class of $q$-ary linear codes with few weights are presented and their weight distributions are determined using Gauss periods. Some of the linear codes obtained are optimal or almost optimal with respect to the Griesmer bound. As s applications, these linear codes can be used to construct secret sharing schemes with nice access structures.
Projective two-weight linear codes are closely related to finite projective spaces and strongly regular graphs. In this paper, a family of $q$-ary projective two-weight linear codes is presented, where $q$ is a power of 2. The parameters of both the codes and their duals are excellent. As applications, the codes are used to derive strongly regular graphs with new parameters and secret sharing schemes with interesting access structures.
Multiply Constant-Weight Codes and the Reliability of Loop Physically Unclonable Functions
我们引入了一类多重等重码，以提高某些物理不可克隆函数（PUF）响应的可靠性。我们扩展了经典的编码方法，从已知的$q$ -ary和等重码构造多重等重码。约翰逊界的类比推导出，并在一定条件下被证明是渐近紧到一个常数因子。我们还研究了乘等重码的速率，有趣的是，证明了这些速率与适当参数的等重码的速率相同。我们的代码建设的渐近分析。
Perfect codes as exact minimizers of quadratic discrepancy in q-ary Hamming spaces
该研究证明q元汉明空间中，当固定长度和基数存在完备码时，完备码是二次偏差的精确极小化元，其存在性等价于显式变分目标的达成。
The weight distributions of some linear codes derived from Kloosterman sums
该研究利用给定定义集构造几类p元少重量线性码，基于有限域上的Kloosterman和确定其重量分布，且证明部分构造的线性码为极小码，可应用于数据存储等领域。
We consider $q$-ary (linear and nonlinear) block codes with exactly two distances: $d$ and $d+δ$. Several combinatorial constructions of optimal such codes are given. In the linear (but not necessary projective) case, we prove that under certain conditions the existence of such linear $2$-weight code with $δ> 1$ implies the following equality of great common divisors: $(d,q) = (δ,q)$. Upper bounds for the maximum cardinality of such codes are derived by linear programming and from few-distance spherical codes. Tables of lower and upper bounds for small $q = 2,3,4$ and $q\,n < 50$ are presented…
Optimal and minimal $p$-ary linear codes from generalized order ideals of hierarchical posets
本文引入分层偏序集的广义序理想，构造了最优和极小的 $p$ 元线性码，确定了补码的重量分布，并获得了违反 Ashikhmin--Barg 条件的无限码族。
A Generic Construction of $q$-ary Near-MDS Codes Supporting 2-Designs with Lengths Beyond $q+1$
本文提出一种通用方法，构造出支持2设计的q-ary近MDS码，其长度超过q+1，结合椭圆曲线码、有限阿贝尔群、子集和与组合设计的新联系，得到无限族码及其重量分布。
A Class of Nonbinary Codes and Their Weight Distribution
本文对偶数$n\geq 4$和任意正整数$k$（其中${\rm gcd}(n/2,k)={\rm gcd}(n/2-k,2k)=d$为奇数），定义了一类$p$ -元码$\mathcal{C}^k$，并完全确定了它们的重量分布，其中$p$为奇素数.作为应用，由这些码构造了一类非二元序列族，并确定了其相关分布。
A q-ary linear code of dimension k is called a maximum weight spectrum (MWS) code if it has the maximum possible number (viz. (q^k-1)/(q-1)) of different non-zero weights. We construct MWS codes from quasi-minimal codes, thus obtaining of much shorter length than hitherto known. By an averaging argument, we show the existence of MWS codes of even shorter length.
Linear Codes with Two or Three Weights From Quadratic Bent Functions
具有较少权值的线性码在秘密共享、认证码、关联方案和强正则图中有应用。本文利用有限域$\gf_p$上的二次Bent函数构造了几类具有两个或三个权的$p$ -元线性码，其中$p$是奇素数.它们包括一些早期的线性码作为特殊情况。这些线性码的重量分布也被确定。
Maximum Distance Separable Codes for Symbol-Pair Read Channels
我们研究了最近由Cassuto和Blaum（2010）引入的符号对读取通道的（符号对）代码。建立了符号对码的Singleton型界，并构造了最优符号对码的无限族。这些码是最大距离可分的（MDS），因为它们满足Singleton型界。与所有已知的q元MDS码的长度为O（q）的经典码相比，我们证明了q元MDS码的符号对码的长度可以为Ω（q^2）。此外，我们完全确定了MDS符号对码的存在某些参数。
Spectrum of Sizes for Perfect Deletion-Correcting Codes
删除校正码的一个特点是，在相同字母表上具有相同长度的完美$t$删除校正码可以具有不同数量的码字，因为半径为$t$的球相对于Levenshte \u <$n距离可以具有不同的大小。因此，在给定长度$n$和字母表大小~ $q$的情况下，确定完美的$t$删除校正码的所有可能大小是有意义的。在本文中，我们完全确定了长度为3的完美$q$ -元1-删除校正码的可能大小的谱，长度为3的所有$q$，和完美$q$ -元2-删除校正码的长度为4的几乎所有$q$，只留下一个小的有限数量的情况下的怀疑。
Linear codes have been an interesting topic in both theory and practice for many years. In this paper, for an odd prime $p$, we present the explicit complete weight enumerator of a family of $p$-ary linear codes constructed with defining set. The weight enumerator is an mmediate result of the complete weight enumerator, which shows that the codes proposed in this paper are three-weight linear codes. Additionally, all nonzero codewords are minimal and thus they are suitable for secret sharing.
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