Complete weight enumerators of a class of linear codes

Let \(\mathbb {F}_{q}\) be the finite field with \(q=p^{m}\) elements, where p is an odd prime and m is a positive integer. For a positive integer t, let \(D\subset \mathbb {F}^{t}_{q}\) and let \({\mathrm {Tr}}_{m}\) be the trace function from \(\mathbb {F}_{q}\) onto \(\mathbb {F}_{p}\). In this paper, let \(D=\{(x_{1},x_{2},\ldots ,x_{t}) \in \mathbb {F}_{q}^{t}\setminus \{(0,0,\ldots ,0)\} : {\mathrm {Tr}}_{m}(x_{1}+x_{2}+\cdots +x_{t})=0\},\) we define a p-ary linear code \(\mathcal {C}_{D}\) by

$$\begin{aligned} \mathcal {C}_{D}=\{\mathbf {c}(a_{1},a_{2},\ldots ,a_{t}) : (a_{1},a_{2},\ldots ,a_{t})\in \mathbb {F}^{t}_{q}\}, \end{aligned}$$

where

$$\begin{aligned} \mathbf {c}(a_{1},a_{2},\ldots ,a_{t})=({\mathrm {Tr}}_{m}(a_{1}x^{2}_{1}+a_{2}x^{2}_{2}+\cdots +a_{t}x^{2}_{t}))_{(x_{1},x_{2},\ldots ,x_{t}) \in D}. \end{aligned}$$

We shall present the complete weight enumerators of the linear codes \(\mathcal {C}_{D}\) and give several classes of linear codes with a few weights. This paper generalizes the results of Yang and Yao (Des Codes Cryptogr, 2016).

     

Complete weight enumerators of a family of three-weight linear codes

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 immediate 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.     

Complete weight enumerators of a class of linear codes with two weights

In this paper, a class of $p$-ary linear codes with two weights is constructed by using the properties of cyclotomic classes of ${\mathbb{F}}_{p^2}^1$. The complete weight enumerators of these linear codes are also determined. In some cases, they are optimal and can be employed to obtain secret sharing schemes with interesting access structures and asymptotically optimal systematic authentication codes.     

Complete weight enumerators of a class of linear codes with two or three weights

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 have at most three weights and they are suitable for applications in secret sharing schemes. This is an extension of the results raised by Wang et al. (2017).     

Weight enumerators of reducible cyclic codes and their dual codes

Let ${\mathbb{F}}_q$ be a finite field and $n$ a positive integer. In this paper, we find a new combinatorial method to determine weight enumerators of reducible cyclic codes and their dual codes of length $n$ over ${\mathbb{F}}_q$, which just generalize results of Zhu et al. (2015); especially, we also give the weight enumerator of a cyclic code, which is viewed as a partial Melas code. Furthermore, weight enumerators obtained in this paper are all in the form of power of a polynomial.     

Complete m-spotty weight enumerators of binary codes,Jacobi forms,and partial Epstein zeta functions

The MacWilliams identity for the complete m-spotty weight enumerators of byte-organized binary codes is a generalization of that for the Hamming weight enumerators of binary codes. In this paper, Jacobi forms are obtained by substituting theta series into the complete m-spotty weight enumerators of binary Type II codes. The Mellin transforms of those theta series provide functional equations for partial Epstein zeta functions which are summands of classical Epstein zeta functions associated with quadratic forms. Then, it is observed that the coefficient matrices appearing in those functional equations are exactly the same as the transformation matrices in the MacWilliams identity for the complete m-spotty weight enumerators of binary self-dual codes.     

On some new m-spotty Lee weight enumerators

One of the objectives of coding theory is to ensure reliability of the computer memory systems that use high-density RAM chips with wide I/O data (e.g. 16, 32, 64 bits). Since these chips are highly vulnerable to m-spotty byte errors, this goal can be achieved using m-spotty byte error-control codes. This paper introduces the m-spotty Lee weight enumerator, the split m-spotty Lee weight enumerator and the joint m-spotty Lee weight enumerator for byte error-control codes over the ring of integers modulo ? (? ≥  2 is an integer) and over arbitrary finite fields, and also discusses some of their applications. In addition, MacWilliams type identities are also derived for these enumerators.     

A formula on the weight distribution of linear codes with applications to AMDS codes

The determination of the weight distribution of linear codes has been a fascinating problem since the very beginning of coding theory. There has been a lot of research on weight enumerators of special cases, such as self-dual codes and codes with small Singleton's defect. We propose a new set of linear relations that must be satisfied by the coefficients of the weight distribution. From these relations we are able to derive known identities (in an easier way) for interesting cases, such as extremal codes, Hermitian codes, MDS and NMDS codes. Moreover, we are able to present for the first time the weight distribution of AMDS codes. We also discuss the link between our results and the Pless equations.     

Weight enumerators of self-orthogonal codes

Canonical forms are given for (i) the weight enumerator of an $\text{|n,}\text{1}\text{2}\text{(n?1)|}$ self-orthogonal code, and (ii) the split weight enumerator (which classifies the codewords according to the weight of the left-and right-half words) of an $\text{|n,}\text{1}\text{2}\text{n|}$ self-dual code.     

Hamming weight enumerators of multi-twisted codes with at most two non-zero constituents

In this paper, we explicitly determine Hamming weight enumerators of several classes of multi-twisted codes over finite fields with at most two non-zero constituents, where each non-zero constituent has dimension 1. Among these classes of multi-twisted codes, we further identify two classes of optimal equidistant linear codes that have nice connections with the theory of combinatorial designs and several other classes of minimal linear codes that are useful in constructing secret sharing schemes with nice access structures. We also illustrate our results with some examples.     

Binary linear codes from vectorial boolean functions and their weight distribution

Binary linear codes with good parameters have important applications in secret sharing schemes, authentication codes, association schemes, and consumer electronics and communications. In this paper, we construct several classes of binary linear codes from vectorial Boolean functions and determine their parameters, by further studying a generic construction developed by Ding et al. recently. First, by employing perfect nonlinear functions and almost bent functions, we obtain several classes of six-weight linear codes which contain the all-one codeword, and determine their weight distribution. Second, we investigate a subcode of any linear code mentioned above and consider its parameters. When the vectorial Boolean function is a perfect nonlinear function or a Gold function in odd dimension, we can completely determine the weight distribution of this subcode. Besides, our linear codes have larger dimensions than the ones by Ding et al.’s generic construction.     

The relative generalized Hamming weight of linear q-ary codes and their subcodes

In this article, some properties of the relative generalized Hamming weight (RGHW) of linear codes and their subcodes are developed with techniques in finite projective geometry. The relative generalized Hamming weights of almost all 4-dimensional q-ary linear codes and their subcodes are determined.      

New quinary linear codes from quasi-twisted codes and their duals

One of the central problems in algebraic coding theory is construction of linear codes with best possible parameters. Quasi-twisted (QT) codes have been promising to solve this problem. Despite extensive search in this class and discovery of a large number of new codes, we have been able to find still more new codes that are QT over the alphabet ${\mathbb{F}}_5$ using a more comprehensive search strategy.     

The structure of linear codes of constant weight

In this paper we determine completely the structure of linear codes over of constant weight. Namely, we determine exactly which modules underlie linear codes of constant weight, and we describe the coordinate functionals involved. The weight functions considered are: Hamming weight, Lee weight, two forms of Euclidean weight, and pre-homogeneous weights. We prove a general uniqueness theorem for virtual linear codes of constant weight. Existence is settled on a case by case basis.

     

The joint weight enumerators and Siegel modular forms

The weight enumerator of a binary doubly even self-dual code is an isobaric polynomial in the two generators of the ring of invariants of a certain group of order 192. The aim of this note is to study the ring of coefficients of that polynomial, both for standard and joint weight enumerators.