In this paper, for an odd prime $p$, by extending Li et al.'s construction \cite{CL2016}, several classes of two-weight and three-weight linear codes over the finite field $\mathbb{F}_p$ are constructed from a defining set, and then their complete weight enumerators are determined by using Weil sums. Furthermore, we show that some examples of these codes are optimal or almost optimal with respect to the Griesmer bound. Our results generalize the corresponding results in \cite{CL2016, GJ2019}.
We construct a class of linear codes by choosing a proper defining set and determine their complete weight enumerators and weight enumerators. The results show that they are at most three-weight codes and they are suitable for applications in secret sharing schemes. We give an extension of the results raised by Wang et al.(2017).
In this paper, based on the theory of defining sets, a class of four-weight or five-weight linear codes over Fp is constructed. The complete weight enumerators of the linear codes are determined by means of Weil sums. In some case, there is an almost optimal code with respect to Griesmer bound, which is also an optimal one according to the online code table. This is an extension of the results raised by Zhang et al.(2020).
In the past few years, linear codes with few weights and their weight analysis have been widely studied. In this paper, we further investigate a class of two-weight or three-weight linear codes from defining sets and determine their weight and complete weight enumerators by application of the theory of quadratic forms and some special Weil sums over finite fields. Some punctured codes of the discussed linear codes are optimal or almost optimal with respect to the Griesmer bound. This paper generalizes some results in \cite{ZhuXu2017,Jian2019}.
It is well-known that Reed-Solomon codes and extended Reed-Solomon codes are two special classes of MDS codes with wide applications in practice. The complete weight enumerators of these codes are very important for determining the capability of both error-detection and error-correction. In this paper, for any positive integer $m$ and prime $p$, basing on the character sums, we determine the complete weight enumerators of the Reed-Solomon code and the extended Reed-Solomon code with dimension $k$ $(k=2,3)$ over $\mathbb{F}_{p^m}$, explictly, which are generalizations of the corresponding resul…
MacWilliams identities for poset level weight enumerators of linear codes
已经考虑了各种度量上的代码，例如Rosenbloom-Tsfasman（RT）、Lee等。最近，偏序集度量上的码得到了研究。偏序集度量是许多度量的一个很好的推广，特别是著名的度量如RT和Hamming度量。偏序集度量可以在具有局部错误发生的信道上实现。已经证明，MacWilliams恒等式对于一般偏序集度量上的码是不可接受的[Kim and Oh，2005]。最近，为了克服这个问题，一些进一步的研究偏序集度量上的MacWilliams恒等式已经提出。本文引入了Frobenius交换环上线性码的偏序集水平权计数元。我们推导出MacWilliams型的身份为每个给定的枚举，在很大程度上推广了以前的文献中讨论的结果。大多数的重量枚举在文献中，如汉明，Rosenbloom-Tsfasman和完整的m-spotty重量枚举遵循这些身份的推论。
In 1997, Shor and Laflamme defined the weight enumerators for quantum error-correcting codes and derived a MacWilliams identity. We extend their work by introducing our double weight enumerators and complete weight enumerators. The MacWilliams identities for these enumerators can be obtained similarly. With the help of MacWilliams identities, we obtain various bounds for asymmetric quantum codes.
We show that one of the Shor-Laflamme weight enumerators of a codeword stabilized quantum code may be interpreted as the distance enumerator of an associated classical code.
We introduce a general class of codes which includes several well-known classes of deletion/insertion correcting codes as special cases. For example, the Helberg code, the Levenshtein code, the Varshamov--Tenengolts code, and most variants of these codes including most of those which have been recently used in studying DNA-based data storage systems are all special cases of our code. Then, using a number theoretic method, we give an explicit formula for the weight enumerator of our code which in turn gives explicit formulas for the weight enumerators and so the sizes of all the aforementioned …
We give restrictions on the weight enumerators of ternary near-extremal self-dual codes of length divisible by $12$ and quaternary near-extremal Hermitian self-dual codes of length divisible by $6$. We consider the weight enumerators for which there is a ternary near-extremal self-dual code of length $12m$ for $m =3,4,5,6$. Also we consider the weight enumerators for which there is a quaternary near-extremal Hermitian self-dual code of length $6m$ for $m =4,5,6$.
The MacWilliams identity for $m$-spotty weight enumerator over $\mathbb{F}_2+u\mathbb{F}_2+\cdots+u^{m-1}\mathbb{F}_2$
在过去的几年中，具有宽I/O数据（例如16、32、64位）的RAM芯片在计算机存储器系统中得到了广泛的使用。这些芯片极易受到一种特殊类型的字节错误的攻击，这种错误称为$m$斑点字节错误，可以使用字节错误控制码有效地检测或纠正这种错误。麦克威廉姆斯恒等式提供了码的重量分布与其对偶码的重量分布之间的关系。本文的主要目的是给出$\mathbbm{F}_{2}+u\mathbbm{F}_{2}+\cdots+u^{m-1}\mathbbm{F}_{2}$上的$m$ -点权枚举数（简称$R_{u, m, 2}$）的MacWilliams恒等式.
The number-theoretic codes are a class of codes defined by single or multiple congruences. These codes are mainly used for correcting insertion and deletion errors, and for correcting asymmetric errors. This paper presents a formula for a generalization of the complete weight enumerator for the number-theoretic codes. This formula allows us to derive the weight enumerators and cardinalities for the number-theoretic codes. As a special case, this paper provides the Hamming weight enumerators and cardinalities of the non-binary Tenengolts' codes, correcting single insertion or deletion. Moreover…
In this note, we give some restrictions on the number of vectors of weight $d/2+1$ in the shadow of a singly even self-dual $[n,n/2,d]$ code. This eliminates some of the possible weight enumerators of singly even self-dual $[n,n/2,d]$ codes for $(n,d)=(62,12)$, $(72,14)$, $(82,16)$, $(90,16)$ and $(100,18)$.
Gleason's 1970 theorem on weight enumerators of self-dual codes has played a crucial role for research in coding theory during the last four decades. Plenty of generalizations have been proved but, to our knowledge, they are all based on the symmetries given by MacWilliams' identities. This paper is intended to be a first step towards a more general investigation of symmetries of weight enumerators. We list the possible groups of symmetries, dealing both with the finite and infinite case, we develop a new algorithm to compute the group of symmetries of a given weight enumerator and apply these…
Periodicity of weight enumerators for codes generated by an integral matrix
Institute of Mathematics for Industry, Kyushu University（九州大学产业数学研究所）；Department of Mathematics, Nagoya Institute of Technology（名古屋工业大学数学系）；Institute for Chemical Reaction Design and Discovery, Hokkaido University（北海道大学化学反应设计与发现研究所）
An explicit formula for a weight enumerator of linear-congruence codes is provided. This extends the work of Bibak and Milenkovic [IEEE ISIT (2018) 431-435] addressing the binary case to the non-binary case. Furthermore, the extension simplifies their proof and provides a complete solution to a problem posed by them.
A class of optimal three-weight cyclic codes of dimension 3 over any finite field was presented by Vega [Finite Fields Appl., 42 (2016) 23-38]. Shortly thereafter, Heng and Yue [IEEE Trans. Inf. Theory, 62(8) (2016) 4501-4513] generalized this result by presenting several classes of cyclic codes with either optimal three weights or a few weights. On the other hand, a class of optimal five-weight cyclic codes of dimension 4 over a prime field was recently presented by Li, et al. [Adv. Math. Commun., 13(1) (2019) 137-156]. One of the purposes of this work is to present a more general description…
In this work, we define a modification of a bordered construction for self-dual codes which utilises $λ$-circulant matrices. We provide the necessary conditions for the construction to produce self-dual codes over finite commutative Frobenius rings of characteristic 2. Using the modified construction together with the neighbour construction, we construct many binary self-dual codes of lengths 54, 68, 82 and 94 with weight enumerators that have previously not been known to exist.
Let H be the standard Hadamard matrix of order two and let K=2^{-1/2}H. It is known that the complete weight enumerator $\ W$ of a binary self-dual code of length $n$ is an eigenvector corresponding to an eigenvalue 1 of the Kronecker power $K^{[n]}.$ For every integer $t$ in the interval [0,n] we define the derivative of order $t$, $W_{<t>},$ of $W$ in such a way that $W_{<t>}$ is in the eigenspace of $\ 1$ of the matrix $K^{[n-t]}.$ For large values of $t,$ $W_{<t>}$ contains less information about the code but has smaller length while $W_{<0>}=W$ completely determines the code. We compute t…
