Linear codes with few weights have applications in secrete sharing, authentication codes, association schemes, and strongly regular graphs. In this paper, several classes of $p$-ary linear codes with two or three weights are constructed from quadratic Bent functions over the finite field $\gf_p$, where $p$ is an odd prime. They include some earlier linear codes as special cases. The weight distributions of these linear codes are also determined.
We give a new simple construction for known binary quadratic symmetric bent and almost bent functions. In particular, for even number of variables, they are self-dual and anti-self-dual quadratic bent functions, respectively, which are not of the Maiorana-McFarland type, but affine equivalent to it.
Inspired by a recent work of Mesnager, we present several new infinite families of quadratic ternary bent, near-bent and 2-plateaued functions from some known quadratic ternary bent functions. Meanwhile, the distribution of the Walsh spectrum of two class of 2-plateaued functions obtained in this paper is completely determined. Additionally, we construct the first class of $p$-ary bent functions of algebraic degree $p$ over the fields of an arbitrary odd characteristic. The proposed class contains non-quadratic $p$-ary bent functions that are affinely inequivalent to known monomial and binomia…
Codebooks with small inner-product correlation have application in unitary space-time modulations, multiple description coding over erasure channels, direct spread code division multiple access communications, compressed sensing, and coding theory. It is interesting to construct codebooks (asymptotically) achieving the Welch bound or the Levenshtein bound. This paper presented a class of generalized bent $\mathbb{Z}_4$-valued quadratic forms, which contain functions of Heng and Yue (Optimal codebooks achieving the Levenshtein bound from generalized bent functions over $\mathbb{Z}_4$. Cryptogr.…
Bent-negabent functions have many important properties for their application in cryptography since they have the flat absolute spectrum under the both Walsh-Hadamard transform and nega-Hadamard transform. In this paper, we present four new systematic constructions of bent-negabent functions on $4k, 8k, 4k+2$ and $8k+2$ variables, respectively, by modifying the truth tables of two classes of quadratic bent-negabent functions with simple form. The algebraic normal forms and duals of these constructed functions are also determined. We further identify necessary and sufficient conditions for those…
Several nonexistence results on generalized bent functions $f:Z_{2}^{n} \rightarrow Z_{m}$ presented by using some knowledge on cyclotomic number fields and their imaginary quadratic subfields.
Linear codes with few weights have applications in consumer electronics, communication, data storage system, secret sharing, authentication codes, association schemes, and strongly regular graphs. This paper first generalizes the method of constructing two-weight and three-weight linear codes of Ding et al. \cite{DD2015} and Zhou et al. \cite{ZLFH2015} to general weakly regular bent functions and determines the weight distributions of these linear codes. It solves the open problem of Ding et al. \cite{DD2015}. Further, this paper constructs new linear codes with two or three weights and presen…
Nonexistence of generalized bent functions and the quadratic norm form equations
Key Laboratory of Cyberspace Security Defense Institute of Information Engineering Chinese Academy of Sciences（中国科学院信息工程研究所网络空间安全防御重点实验室）；Beijing Key Laboratory of Security and Privacy in Intelligent Transportation School of Cyberspace Science and Technology Beijing Jiaotong University（北京交通大学网络空间科学与技术学院智能交通安全与隐私北京市重点实验室）
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…
On quadratic binomial vectorial functions with maximal bent components
Anhui Agricultural University（安徽农业大学）；University of Science and Technology of China（中国科学技术大学）；Hefei National Laboratory（合肥国家实验室）；Hefei University of Technology（合肥工业大学）；State Key Laboratory of Cyberspace Security Defense, Institute of Information Engineering, Chinese Academy of Sciences（中国科学院信息工程研究所网络空间安全防御国家重点实验室）
The minimal Hamming distance between distinct $p$-ary bent functions of $2n$ variables is proved to be $p^n$ for any prime $p$. It is shown that the number of $p$-ary bent functions at the distance $p^n$ from the quadratic bent function is equal to $p^n(p^{n-1}+1)\cdots(p+1)(p-1)$ as $p>2$.
Bent functions as optimal combinatorial objects are difficult to characterize and construct. In the literature, bent idempotents are a special class of bent functions and few constructions have been presented, which are restricted by the degree of finite fields and have algebraic degree no more than 4. In this paper, several new infinite families of bent functions are obtained by adding the the algebraic combination of linear functions to some known bent functions and their duals are calculated. These bent functions contain some previous work on infinite families of bent functions by Mesnager …
In this paper, several classes of three-weight ternary linear codes from non-weakly regular dual-bent functions are constructed based on a generic construction method. Instead of the whole space, we use the subspaces $B_+(f)$ or $B_-(f)$ associated with a ternary non-weakly regular dual-bent function $f$. Unusually, we use the pre-image sets of the dual function $f^*$ in $B_+(f)$ or $B_-(f)$ as the defining sets of the corresponding codes. Since the size of the defining sets of the constructed codes is flexible, it enables us to construct several codes with different parameters for a fixed dim…
A Type IV-II Z4-code is a self-dual code over Z4 with the property that all Euclidean weights are divisible by eight and all codewords have even Hamming weight. In this paper we use generalized bent functions for a construction of self-orthogonal codes over Z4 of length $2^m$, for $m$ odd, $m \geq 3$, and prove that for $m \geq 5$ those codes can be extended to Type IV-II Z4-codes. From that family of Type IV-II Z4-codes, we obtain a family of self-dual Type II binary codes by using Gray map. We also consider the weight distributions of the obtained codes and the structure of the supports of t…
Bent functions $f: V_{n}\rightarrow \mathbb{F}_{p}$ with certain additional properties play an important role in constructing partial difference sets, where $V_{n}$ denotes an $n$-dimensional vector space over $\mathbb{F}_{p}$, $p$ is an odd prime. In \cite{Cesmelioglu1,Cesmelioglu2}, the so-called vectorial dual-bent functions are considered to construct partial difference sets. In \cite{Cesmelioglu1}, Çeşmelioǧlu \emph{et al.} showed that for vectorial dual-bent functions $F: V_{n}\rightarrow V_{s}$ with certain additional properties, the preimage set of $0$ for $F$ forms a partial differenc…
In this paper, several new classes of Boolean functions with few Walsh transform values, including bent, semi-bent and five-valued functions, are obtained by adding the product of two or three linear functions to some known bent functions.Numerical results show that the proposed class contains cubic bent functions that are affinely inequivalent to all known quadratic ones. Meanwhile, we determine the distribution of the Walsh spectrum of five-valued functions constructed in this paper.
We obtain the following results. For any prime $q$ the minimal Hamming distance between distinct regular $q$-ary bent functions of $2n$ variables is equal to $q^n$. The number of $q$-ary regular bent functions at the distance $q^n$ from the quadratic bent function $Q_n=x_1x_2+\dots+x_{2n-1}x_{2n}$ is equal to $q^n(q^{n-1}+1)\cdots(q+1)(q-1)$ for $q>2$. The Hamming distance between distinct binary $s$-plateaued functions of $n$ variables is not less than $2^{\frac{s+n-2}{2}}$ and the Hamming distance between distinctternary $s$-plateaued functions of $n$ variables is not less than $3^{\frac{s+n…
In this paper we define the (edge-weighted) Cayley graph associated to a generalized Boolean function, introduce a notion of strong regularity and give several of its properties. We show some connections between this concept and generalized bent functions (gbent), that is, functions with flat Walsh-Hadamard spectrum. In particular, we find a complete characterization of quartic gbent functions in terms of the strong regularity of their associated Cayley graph.
It has been an active research issue for many years to construct new bent functions. For $k$ odd with $\gcd(n, k)=1$, and $a\in\mathbb{F}_{3^n}^{*}$, the function $f(x)=Tr(ax^{\frac{3^k+1}{2}})$ is weakly regular bent over $\mathbb{F}_{3^n}$, where $Tr(\cdot):\mathbb{F}_{3^n}\rightarrow\mathbb{F}_3$ is the trace function. This is the well-known Coulter-Matthews bent function. In this paper, we determine the dual function of $f(x)$ completely. As a consequence, we find many classes of ternary bent functions not reported in the literature previously. Such bent functions are not quadratic if $k>1…
In 2018, Pott, at al. have studied in [IEEE Transactions on Information Theory. Volume: 64, Issue: 1, 2018] the maximum number of bent components of vectorial function. They have presented serval nice results and suggested several open problems in this context. This paper is in the continuation of their study in which we solve two open problems raised by Pott et al. and partially solve an open problem raised by the same authors. Firstly, we prove that for a vectorial function, the property of having the maximum number of bent components is invariant under the so-called CCZ equivalence. Secondl…
We obtain an explicit criterion for $p$-ary functions to produce association schemes in terms of their Walsh spectrum. Employing this characterization, we explicitly find a correlation between $p$-ary bent functions and association schemes; to be more exact, we prove that a $p$-ary bent function induces a $p$-class association scheme if and only if the function is weakly regular. As applications of our main criterion, we construct many infinite families of few-class association schemes arising from $p$-ary functions. Furthermore, we present four classes of $p$-ary two-weight linear codes, whic…
Linear codes with few weights have been an interesting subject of study for many years, as these codes have applications in secrete sharing, authentication codes, association schemes, and strongly regular graphs. In this paper, linear codes with a few weights are constructed from inhomogeneous quadratic functions over the finite field $\gf(p)$, where $p$ is an odd prime. They include some earlier linear codes as special cases. The weight distributions of these linear codes are also determined.
We restrict a degenerate quadratic form $f$ over a finite field of odd characteristic to subspaces. Thus, a quotient space related to $f$ is introduced. Then we get a non-degenerate quadratic form induced by $f$ over the quotient space. Some related results on the subspaces and quotient space are obtained. Based on this, we solve the weight hierarchies of a family of linear codes related to $f.$
The generalized Hamming weight of linear codes is a natural generalization of the minimum Hamming distance. They convey the structural information of a linear code and determine its performance in various applications, and have become one of important research topics in coding theory. Recently, Li (IEEE Trans. Inf. Theory, 67(1): 124-129, 2021) and Li and Li (Discrete Math., 345: 112718, 2022) obtained the complete weight hierarchy of linear codes from a quadratic form over a finite field of odd characteristic by analysis of the solutions of the restricted quadratic equation in its subspace. I…
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 \emph{et al.} recently. First, by employing perfect nonlinear functions and almost bent functions, we obtain several classes of six-weight linear codes which contains the all-one codeword. Second, we investigate a subcode o…
In this paper, we begin by reviewing some of the known properties of QQR codes and proved that $PSL_2(p)$ acts on the extended QQR code when $p \equiv 3 \pmod 4$. Using this discovery, we then showed their weight polynomials satisfy a strong divisibility condition, namely that they are divisible by $(x^2 + y^2)^{d-1}$, where $d$ is the corresponding minimum distance. Using this result, we were able to construct an efficient algorithm to compute weight polynomials for QQR codes and correct errors in existing results on quadratic residue codes. In the second half, we use the relation between the…
In this paper, we discuss the generalized Hamming weights of a class of linear codes associated with non-degenerate quadratic forms. In order to do so, we study the quadratic forms over subspaces of finite field and obtain some interesting results about subspaces and their dual spaces. On this basis, we solve all the generalized Hamming weights of these linear codes.
In this paper, we present a construction of linear codes over $\f_{2^t}$ from Boolean functions, which is a generalization of Ding's method \cite[Theorem 9]{Ding15}. Based on this construction, we give two classes of linear codes $\tilde{\C}_{f}$ and $\C_f$ (see Theorem \ref{thm-maincode1} and Theorem \ref{thm-maincodenew}) over $\f_{2^t}$ from a Boolean function $f:\f_{q}\rightarrow \f_2$, where $q=2^n$ and $\f_{2^t}$ is some subfield of $\f_{q}$. The complete weight enumerator of $\tilde{\C}_{f}$ can be easily determined from the Walsh spectrum of $f$, while the weight distribution of the co…
Let $f_n(x_0, x_1, \ldots, x_{n-1})$ denote the algebraic normal form (polynomial form) of a rotation symmetric (RS) Boolean function of degree $d$ in $n \geq d$ variables and let $wt(f_n)$ denote the Hamming weight of this function. Let $(0, a_1, \ldots, a_{d-1})_n$ denote the function $f_n$ of degree $d$ in $n$ variables generated by the monomial $x_0x_{a_1} \cdots x_{a_{d-1}}.$ Such a function $f_n$ is called monomial rotation symmetric (MRS). It was proved in a $2012$ paper that for any MRS $f_n$ with $d=3,$ the sequence of weights $\{w_k = wt(f_k):~k = 3, 4, \ldots\}$ satisfies a homogene…
