The linear codes with a few weights have been applied widely in combinatorial designs, secret sharing, association schemes, authentication codes and strongly regular graphs. In this paper, we first correct an erroneous result about the exponential sum $\sum_{x\in \mathbb{F}_{2^{e}}}χ_1\left(ax^{2^α+1}+bx\right).$ Then, using the above exponential sum, we construct several families of binary linear codes of $3$-weight and determine their weight distributions. Moreover, Most of them can be used in secret sharing schemes.
Correlation measure of order $k$ is an important measure of randomness in binary sequences. This measure tries to look for dependence between several shifted version of a sequence. We study the relation between the correlation measure of order $k$ and another two pseudorandom measures: the $N$th linear complexity and the $N$th maximum order complexity. We simplify and improve several state-of-the-art lower bounds for these two measures using the Hamming bound as well as weaker bounds derived from it.
Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. Inspired by the recent work on binary cyclic codes published in IEEE Trans. Inf. Theory, vol. 68, no. 12, pp. 7842-7849, 2022, the objectives of this paper are the construction and analyses of five infinite families of binary cyclic codes with parameters $[n, k]$ and $(n-6)/3 \leq k \leq 2(n+6)/3$. Three of the five families of binary cyclic codes and their duals have a very good lower bound on their minimum distances and …
Binary duadic codes are an interesting subclass of cyclic codes since they have large dimensions and their minimum distances may have a square-root bound. In this paper, we present several families of binary duadic codes of length $2^m-1$ and develop some lower bounds on their minimum distances by using the BCH bound on cyclic codes, which partially solves one case of the open problem proposed in \cite{LLD}. It is shown that the lower bounds on their minimum distances are close to the square root bound. Moreover, the parameters of the dual and extended codes of these binary duadic codes are in…
Two general constructions of linear codes with functions over finite fields have been extensively studied in the literature. The first one is given by $\mathcal{C}(f)=\left\{ {\rm Tr}(af(x)+bx)_{x \in \mathbb{F}_{q^m}^*}: a,b \in \mathbb{F}_{q^m} \right\}$, where $q$ is a prime power, $\bF_{q^m}^*=\bF_{q^m} \setminus \{0\}$, $\tr$ is the trace function from $\bF_{q^m}$ to $\bF_q$, and $f(x)$ is a function from $\mathbb{F}_{q^m}$ to $\mathbb{F}_{q^m}$ with $f(0)=0$. Almost bent functions, quadratic functions and some monomials on $\bF_{2^m}$ were used in the first construction, and many familie…
The merit factor of binary arrays derived from the quadratic character
本文计算了由二次特征标导出的两类二元二维阵列在行和列的所有循环旋转下的渐近优值因子。这些族中的数组大小为p x q，其中p和q不一定是不同的奇素数，并且可以被认为是勒让德序列的二维推广。这两个家族的价值因子的渐近值通常是不同的，虽然最大渐近价值因子，采取了所有循环旋转的行和列，等于36/13的两个家庭。这是第一个非平凡的理论结果的渐近价值因子的家庭真正的二维二元阵列。
Diameter perfect codes form a natural generalization for perfect codes. They are based on the code-anticode bound which generalizes the sphere-packing bound. The code-anticode bound was proved by Delsarte for distance-regular graphs and it holds for some other metrics too. In this paper we prove the bound for non-binary constant-weight codes with the Hamming metric and characterize the diameter perfect codes and the maximum size anticodes for these codes. We distinguish between six families of non-binary diameter constant-weight codes and four families of maximum size non-binary constant-weigh…
Combinatorial $t$-designs have been an important research subject for many years, as they have wide applications in coding theory, cryptography, communications and statistics. The interplay between coding theory and $t$-designs has been attracted a lot of attention for both directions. It is well known that a linear code over any finite field can be derived from the incidence matrix of a $t$-design, meanwhile, that the supports of all codewords with a fixed weight in a code also may hold a $t$-design. In this paper, by determining the weight distribution of a class of linear codes derived from…
Linear codes with a few weights can be applied to communication, consumer electronics and data storage system. In addition, the weight hierarchy of linear codes has many applications such as on the type II wire-tap channel, dealing with $t$-resilient functions and trellis or branch complexity of linear codes and so on. In this paper, we first present a formula for computing the weight hierarchies of linear codes constructed by the generalized method of defining sets. Then, we construct two classes of binary linear codes with a few weights and determine their weight distributions and weight hie…
This paper provides a combinatorial characterization of weak algebraic manipulation detection (AMD) codes via a kind of generalized external difference families called bounded standard weighted external difference families (BSWEDFs). By means of this characterization, we improve a known lower bound on the maximum probability of successful tampering for the adversary's all possible strategies in weak AMD codes. We clarify the relationship between weak AMD codes and BSWEDFs with various properties. We also propose several explicit constructions for BSWEDFs, some of which can generate new optimal…
Combinatorial $t$-designs have nice applications in coding theory, finite geometries and several engineering areas. The objective of this paper is to study how to obtain $3$-designs with $2$-transitive permutation groups. The incidence structure formed by the orbits of a base block under the action of the general affine groups, which are $2$-transitive, is considered. A characterization of such incidence structure to be a $3$-design is presented, and a sufficient condition for the stabilizer of a base block to be trivial is given. With these general results, infinite families of $3$-designs ar…
A statistical estimation algorithm of the weight distribution of a linear code is shown, based on using its generator matrix as a compression function on random bit strings.
Boolean functions have very nice applications in cryptography and coding theory, which have led to a lot of research focusing on their applications. The objective of this paper is to construct binary linear codes with few weights from the defining set, which is defined by some special Boolean functions and some additional restrictions. First, we provide two general constructions of binary linear codes with three or four weights from Boolean functions with at most three Walsh transform values and determine the parameters of their dual codes. Then many classes of binary linear codes with explici…
In this paper, several families of irreducible constacyclic codes over finite fields and their duals are studied. The weight distributions of these irreducible constacyclic codes and the parameters of their duals are settled. Several families of irreducible constacyclic codes with a few weights and several families of optimal constacyclic codes are constructed. As by-products, a family of $[2n, (n-1)/2, d \geq 2(\sqrt{n}+1)]$ irreducible cyclic codes over $\gf(q)$ and a family of $[(q-1)n, (n-1)/2, d \geq (q-1)(\sqrt{n}+1)]$ irreducible cyclic codes over $\gf(q)$ are presented, where $n$ is a …
Subfield codes of linear codes over finite fields have recently received much attention. Some of these codes are optimal and have applications in secrete sharing, authentication codes and association schemes. In this paper, the $q$-ary subfield codes $C_{f,g}^{(q)}$ of six different families of linear codes $C_{f,g}$ parametrized by two functions $f, g$ over a finite field $F_{q^m}$ are considered and studied, respectively. The parameters and (Hamming) weight distribution of $C_{f,g}^{(q)}$ and their punctured codes $\bar{C}_{f,g}^{(q)}$ are explicitly determined. The parameters of the duals o…
Key Laboratory of Intelligent Computing Signal Processing, Ministry of Education, School of Mathematical Sciences, Anhui University（安徽大学数学科学学院）；State Key Laboratory of Integrated Service Networks, Xidian University（西安电子科技大学集成服务网络国家重点实验室）；I2M (Aix Marseille Univ, CNRS)（艾克斯马赛大学CNRS数学研究所）
Boolean functions can be used to construct binary linear codes in many ways, and vice versa. The objective of this short article is to point out a connection between the weight distributions of all projective binary linear codes and the Walsh spectra of all Boolean functions. New research problems are also proposed.
Motivated by applications in polymer-based data storage we introduced the new problem of characterizing the code rate and designing constant-weight binary $B_2$-sequences. Binary $B_2$-sequences are collections of binary strings of length $n$ with the property that the real-valued sums of all distinct pairs of strings are distinct. In addition to this defining property, constant-weight binary $B_2$-sequences also satisfy the constraint that each string has a fixed, relatively small weight $ω$ that scales linearly with $n$. The constant-weight constraint ensures low-cost synthesis and uniform p…
We introduce the concept of spread of a code, and we specialize it to the case of maximum weight spectrum (MWS) codes. We classify two newly-defined sub-families of MWS codes according to their weight distributions, and completely describe their fundamental parameters. We focus on one of these families, the strictly compact MWS codes, proving their optimality as MWS codes and linking them to known codes.
In this work, we explore the relationship between free resolution of some monomial ideals and Generalized Hamming Weights (GHWs) of binary codes. More precisely, we look for a structure smaller than the set of codewords of minimal support that provides us some information about the GHWs. We prove that the first and second generalized Hamming weight of a binary linear code can be computed (by means of a graded free resolution) from a set of monomials associated to a binomial ideal related with the code. Moreover, the remaining weights are bounded by the Betti numbers for that set.
Linear complementary dual (LCD) codes are linear codes that intersect with their dual codes trivially. We study the largest minimum weight $d_2(n,k)$ among all binary LCD $[n,k]$ codes and the largest minimum weight $d_3(n,k)$ among all ternary LCD $[n,k]$ codes. The largest minimum weights $d_2(n,5)$ and $d_3(n,4)$ are partially determined. We also determine the largest minimum weights $d_2(n,n-5)$, $d_3(n,n-i)$ for $i \in \{2,3,4\}$, and $d_3(n,k)$ for $n \in \{11,12,\ldots,19\}$.
