arXiv subjects

Chunming Tang

Publications and source records attributed to Chunming Tang.

At least 37 records · Page 2Linked to original sources

arXiv2023

A Partially Feasible Distributed SQO Method for Two-block General Linearly Constrained Smooth Optimization

This paper discusses a class of two-block smooth large-scale optimization problems with both linear equality and linear inequality constraints, which have a wide range of applications, such as economic power dispatch, data mining, signal processing, etc.Our goal is to develop a novel partially feasible distributed (PFD) sequential quadratic optimization (SQO) method (PFD-SQO method) for this kind of problems. The design of the method is based on the ideas of SQO method and augmented Lagrangian Jacobian splitting scheme as well as feasible direction method,which decomposes the quadratic optimization (QO) subproblem into two small-scale QOs that can be solved independently and parallelly. A novel disturbance contraction term that can be suitably adjusted is introduced into the inequality constraints so that the feasible step size along the search direction can be increased to 1. The new iteration points are generated by the Armijo line search and the partially augmented Lagrangian function that only contains equality constraints as the merit function. The iteration points always satisfy all the inequality constraints of the problem. The theoretical properties, such as global convergence, iterative complexity, superlinear and quadratic rates of convergence of the proposed PFD-SQO method are analyzed under appropriate assumptions, respectively. Finally, the numerical effectiveness of the method is tested on a class of academic examples and an economic power dispatch problem, which shows that the proposed method is quite promising.

math.OC↗

arXiv2023

Codes and Pseudo-Geometric Designs from the Ternary $m$-Sequences with Welch-type decimation $d=2\cdot 3^{(n-1)/2}+1$

Pseudo-geometric designs are combinatorial designs which share the same parameters as a finite geometry design, but which are not isomorphic to that design. As far as we know, many pseudo-geometric designs have been constructed by the methods of finite geometries and combinatorics. However, none of pseudo-geometric designs with the parameters $S\left (2, q+1,(q^n-1)/(q-1)\right )$ is constructed by the approach of coding theory. In this paper, we use cyclic codes to construct pseudo-geometric designs. We firstly present a family of ternary cyclic codes from the $m$-sequences with Welch-type decimation $d=2\cdot 3^{(n-1)/2}+1$, and obtain some infinite family of 2-designs and a family of Steiner systems $S\left (2, 4, (3^n-1)/2\right )$ using these cyclic codes and their duals. Moreover, the parameters of these cyclic codes and their shortened codes are also determined. Some of those ternary codes are optimal or almost optimal. Finally, we show that one of these obtained Steiner systems is inequivalent to the point-line design of the projective space $\mathrm{PG}(n-1,3)$ and thus is a pseudo-geometric design.

cs.IT↗

arXiv2023

Adjoints of Matroids

We show that an adjoint of a loopless matroid is connected if and only if it itself is connected. Our first goal is to study the adjoint of modular matroids. We prove that a modular matroid has only one adjoint (up to isomorphism) which can be given by its opposite lattice, and proceed to present some alternative characterizations of modular matroids associated to adjoints and opposite lattices. The other purpose is to investigate the adjoint sequence $ad^0M,adM,ad^2M,\ldots$ of a connected matroid $M$. We classify such adjoint sequences into three types: finite, cyclic and convergent. For the first two types, the adjoint sequences eventually stabilize at the finite projective geometries except for free matroids. For the last type, the infinite non-repeating adjoint sequences are convergent to the infinite projective geometries.

math.CO↗

arXiv2023

On Constructions and Enumeration of Vectorial Hyper-bent Functions in the $\cP\cS_{ap}^{\#}$ Class

The purpose of this paper is to give explicit constructions of vectorial hyper-bent functions in the $\cP\cS_{ap}^{\#}$ class. It seems that the explicit constructions were so far known only for very special cases. To this end, we present a sufficient and necessary condition of this family of vectorial functions to be hyper-bent. The conditions are expressed in terms of group ring. Using this characterization, explicit constructions of vectorial hyper-bent functions of the $\cP\cS_{ap}^{\#}$ class via balanced functions are proposed. Furthermore, exact number of vectorial hyper-bent functions in the $\cP\cS_{ap}^{\#}$ class is found. The results improve some previous work. Moreover, we solve a problem of counting vectorial hyper-bent functions left by Muratović-Ribić, Pasalic and Ribić in [{\em IEEE Trans. Inform. Theory}, 60 (2014), pp. 4408-4413].

math.CO↗

arXiv2022

Some $3$-designs and shortened codes from binary cyclic codes with three zeros

Linear codes and $t$-designs are interactive with each other. It is well known that some $t$-designs have been constructed by using certain linear codes in recent years. However, only a small number of infinite families of the extended codes of linear codes holding an infinite family of $t$-designs with $t\geq 3$ are reported in the literature. In this paper, we study the extended codes of the augmented codes of a class of binary cyclic codes with three zeros and their dual codes, and show that those codes hold $3$-designs. Furthermore, we obtain some shortened codes from the studied cyclic codes and explicitly determine their parameters. Some of those shortened codes are optimal or almost optimal.

cs.IT↗

arXiv2022

Infinite families of cyclic and negacyclic codes supporting 3-designs

Interplay between coding theory and combinatorial $t$-designs has been a hot topic for many years for combinatorialists and coding theorists. Some infinite families of cyclic codes supporting infinite families of $3$-designs have been constructed in the past 50 years. However, no infinite family of negacyclic codes supporting an infinite family of $3$-designs has been reported in the literature. This is the main motivation of this paper. Let $q=p^m$, where $p$ is an odd prime and $m \geq 2$ is an integer. The objective of this paper is to present an infinite family of cyclic codes over $\gf(q)$ supporting an infinite family of $3$-designs and two infinite families of negacyclic codes over $\gf(q^2)$ supporting two infinite families of $3$-designs. The parameters and the weight distributions of these codes are determined. The subfield subcodes of these negacyclic codes over $\gf(q)$ are studied. Three infinite families of almost MDS codes are also presented. A constacyclic code over GF($4$) supporting a $4$-design and six open problems are also presented in this paper.

cs.IT↗

arXiv2022

A class of twisted generalized Reed-Solomon codes

Let $\mathbb{F}_q$ be a finite field of size $q$ and $\mathbb{F}_q^*$ the set of non-zero elements of $\mathbb{F}_q$. In this paper, we study a class of twisted generalized Reed-Solomon code $C_\ell(D, k, η, \vec{v})\subset \mathbb{F}_q^n$ generated by the following matrix \[ \left(\begin{array}{cccc} v_{1} & v_{2} & \cdots & v_{n} \\ v_{1} α_{1} & v_{2} α_{2} & \cdots & v_{n} α_{n} \\ \vdots & \vdots & \ddots & \vdots \\ v_{1} α_{1}^{\ell-1} & v_{2} α_{2}^{\ell-1} & \cdots & v_{n} α_{n}^{\ell-1} \\ v_{1} α_{1}^{\ell+1} & v_{2} α_{2}^{\ell+1} & \cdots & v_{n} α_{n}^{\ell+1} \\ \vdots & \vdots & \ddots & \vdots \\ v_{1} α_{1}^{k-1} & v_{2} α_{2}^{k-1} & \cdots & v_{n} α_{n}^{k-1} \\ v_{1}\left(α_{1}^{\ell}+ηα_{1}^{q-{2}}\right) & v_{2}\left(α_{2}^{\ell}+ ηα_{2}^{q-2}\right) &\cdots & v_{n}\left(α_{n}^{\ell}+ηα_{n}^{q-2}\right) \end{array}\right) \] where $0\leq \ell\leq k-1,$ the evaluation set $D=\{α_{1},α_{2},\cdots, α_{n}\}\subseteq \mathbb{F}_q^*$, scaling vector $\vec{v}=(v_1,v_2,\cdots,v_n)\in (\mathbb{F}_q^*)^n$ and $η\in\mathbb{F}_q^*$. The minimum distance and dual code of $C_\ell(D, k, η, \vec{v})$ will be determined. For the special case $\ell=k-1,$ a sufficient and necessary condition for $C_{k-1}(D, k, η, \vec{v})$ to be self-dual will be given. We will also show that the code is MDS or near-MDS. Moreover, a complete classification when the code is near-MDS or MDS will be presented.

cs.IT↗

arXiv2021

An infinite family of antiprimitive cyclic codes supporting Steiner systems $S(3,8, 7^m+1)$

Coding theory and combinatorial $t$-designs have close connections and interesting interplay. One of the major approaches to the construction of combinatorial t-designs is the employment of error-correcting codes. As we all known, some $t$-designs have been constructed with this approach by using certain linear codes in recent years. However, only a few infinite families of cyclic codes holding an infinite family of $3$-designs are reported in the literature. The objective of this paper is to study an infinite family of cyclic codes and determine their parameters. By the parameters of these codes and their dual, some infinite family of $3$-designs are presented and their parameters are also explicitly determined. In particular, the complements of the supports of the minimum weight codewords in the studied cyclic code form a Steiner system. Furthermore, we show that the infinite family of cyclic codes admit $3$-transitive automorphism groups.

cs.IT↗

arXiv2021

An Open Problem on the Bentness of Mesnager's Functions

Let $n=2m$. In the present paper, we study the binomial Boolean functions of the form $$f_{a,b}(x) = \mathrm{Tr}_1^{n}(a x^{2^m-1 }) +\mathrm{Tr}_1^{2}(bx^{\frac{2^n-1}{3} }), $$ where $m$ is an even positive integer, $a\in \mathbb{F}_{2^n}^*$ and $b\in \mathbb{F}_4^*$. We show that $ f_{a,b}$ is a bent function if the Kloosterman sum $$K_{m}\left(a^{2^m+1}\right)=1+ \sum_{x\in \mathbb{F}_{2^m}^*} (-1)^{\mathrm{Tr}_1^{m}(a^{2^m+1} x+ \frac{1}{x})}$$ equals $4$, thus settling an open problem of Mesnager. The proof employs tools including computing Walsh coefficients of Boolean functions via multiplicative characters, divisibility properties of Gauss sums, and graph theory.

cs.IT↗

arXiv2021

On Infinite Families of Narrow-Sense Antiprimitive BCH Codes Admitting 3-Transitive Automorphism Groups and their Consequences

The Bose-Chaudhuri-Hocquenghem (BCH) codes are a well-studied subclass of cyclic codes that have found numerous applications in error correction and notably in quantum information processing. A subclass of attractive BCH codes is the narrow-sense BCH codes over the Galois field $\mathrm{GF}(q)$ with length $q+1$, which are closely related to the action of the projective general linear group of degree two on the projective line. This paper aims to study some of the codes within this class and specifically narrow-sense antiprimitive BCH codes (these codes are also linear complementary duals (LCD) codes that have interesting practical recent applications in cryptography, among other benefits). We shall use tools and combine arguments from algebraic coding theory, combinatorial designs, and group theory (group actions, representation theory of finite groups, etc.) to investigate narrow-sense antiprimitive BCH Codes and extend results from the recent literature. Notably, the dimension, the minimum distance of some $q$-ary BCH codes with length $q+1$, and their duals are determined in this paper. The dual codes of the narrow-sense antiprimitive BCH codes derived in this paper include almost MDS codes. Furthermore, the classification of $\mathrm{PGL} (2, p^m)$-invariant codes over $\mathrm{GF} (p^h)$ is completed. As an application of this result, the $p$-ranks of all incidence structures invariant under the projective general linear group $\mathrm{ PGL }(2, p^m)$ are determined. Furthermore, infinite families of narrow-sense BCH codes admitting a $3$-transitive automorphism group are obtained. Via these BCH codes, a coding-theory approach to constructing the Witt spherical geometry designs is presented. The BCH codes proposed in this paper are good candidates for permutation decoding, as they have a relatively large group of automorphisms.

cs.IT↗

arXiv2021

The subfield codes and subfield subcodes of a family of MDS codes

Maximum distance separable (MDS) codes are very important in both theory and practice. There is a classical construction of a family of $[2^m+1, 2u-1, 2^m-2u+3]$ MDS codes for $1 \leq u \leq 2^{m-1}$, which are cyclic, reversible and BCH codes over $\mathrm{GF}(2^m)$. The objective of this paper is to study the quaternary subfield subcodes and quaternary subfield codes of a subfamily of the MDS codes for even $m$. A family of quaternary cyclic codes is obtained. These quaternary codes are distance-optimal in some cases and very good in general. Furthermore, infinite families of $3$-designs from these quaternary codes are presented.

cs.IT↗

arXiv2020

The Projective General Linear Group $\mathrm{PGL}_2(\mathrm{GF}(2^m))$ and Linear Codes of Length $2^m+1$

The projective general linear group $\mathrm{PGL}_2(\mathrm{GF}(2^m))$ acts as a $3$-transitive permutation group on the set of points of the projective line. The first objective of this paper is to prove that all linear codes over $\mathrm{GF}(2^h)$ that are invariant under $\mathrm{PGL}_2(\mathrm{GF}(2^m))$ are trivial codes: the repetition code, the whole space $\mathrm{GF}(2^h)^{2^m+1}$, and their dual codes. As an application of this result, the $2$-ranks of the (0,1)-incidence matrices of all $3$-$(q+1,k,λ)$ designs that are invariant under $\mathrm{PGL}_2(\mathrm{GF}(2^m))$ are determined. The second objective is to present two infinite families of cyclic codes over $\mathrm{GF}(2^m)$ such that the set of the supports of all codewords of any fixed nonzero weight is invariant under $\mathrm{PGL}_2(\mathrm{GF}(2^m))$, therefore, the codewords of any nonzero weight support a 3-design. A code from the first family has parameters $[q+1,q-3,4]_q$, where $q=2^m$, and $m\ge 4$ is even. The exact number of the codewords of minimum weight is determined, and the codewords of minimum weight support a 3-$(q+1,4,2)$ design. A code from the second family has parameters $[q+1,4,q-4]_q$, $q=2^m$, $m\ge 4$ even, and the minimum weight codewords support a 3-$(q +1,q-4,(q-4)(q-5)(q-6)/60)$ design, whose complementary 3-$(q +1, 5, 1)$ design is isomorphic to the Witt spherical geometry with these parameters. A lower bound on the dimension of a linear code over $\mathrm{GF}(q)$ that can support a 3-$(q +1,q-4,(q-4)(q-5)(q-6)/60)$ design is proved, and it is shown that the designs supported by the codewords of minimum weight in the codes from the second family of codes meet this bound.

cs.IT↗

arXiv2020

Shortened linear codes from APN and PN functions

Linear codes generated by component functions of perfect nonlinear (PN) and almost perfect nonlinear (APN) functions and the first-order Reed-Muller codes have been an object of intensive study in coding theory. The objective of this paper is to investigate some binary shortened codes of two families of linear codes from APN functions and some $p$-ary shortened codes associated with PN functions. The weight distributions of these shortened codes and the parameters of their duals are determined. The parameters of these binary codes and $p$-ary codes are flexible. Many of the codes presented in this paper are optimal or almost optimal. The results of this paper show that the shortening technique is very promising for constructing good codes.

cs.IT↗

arXiv2020

The linear codes of t-designs held in the Reed-Muller and Simplex codes

A fascinating topic of combinatorics is $t$-designs, which have a very long history. The incidence matrix of a $t$-design generates a linear code over GF$(q)$ for any prime power $q$, which is called the linear code of the $t$-design over GF$(q)$. On the other hand, some linear codes hold $t$-designs for some $t \geq 1$. The purpose of this paper is to study the linear codes of some $t$-designs held in the Reed-Muller and Simplex codes. Some general theory for the linear codes of $t$-designs held in linear codes is presented. Open problems are also presented.

cs.IT↗

arXiv2020

Shortened Linear Codes over Finite Fields

The puncturing and shortening technique are two important approaches to constructing new linear codes from old ones. In the past 70 years, a lot of progress on the puncturing technique has been made, and many works on punctured linear codes have been done. Many families of linear codes with interesting parameters have been obtained with the puncturing technique. However, little research on the shortening technique has been done and there are only a handful references on shortened linear codes. The first objective of this paper is to prove some general theory for shortened linear codes. The second objective is to study some shortened codes of the Hamming codes, Simplex codes, some Reed-Muller codes, and ovoid codes. Eleven families of optimal shortened codes with interesting parameters are presented in this paper. As a byproduct, five infinite families of $2$-designs are also constructed from some of the shortened codes presented in this paper.

cs.IT↗

arXiv2020

Fast algebraic immunity of Boolean functions and LCD codes

Nowadays, the resistance against algebraic attacks and fast algebraic attacks are considered as an important cryptographic property for Boolean functions used in stream ciphers. Both attacks are very powerful analysis concepts and can be applied to symmetric cryptographic algorithms used in stream ciphers. The notion of algebraic immunity has received wide attention since it is a powerful tool to measure the resistance of a Boolean function to standard algebraic attacks. Nevertheless, an algebraic tool to handle the resistance to fast algebraic attacks is not clearly identified in the literature. In the current paper, we propose a new parameter to measure the resistance of a Boolean function to fast algebraic attack. We also introduce the notion of fast immunity profile and show that it informs both on the resistance to standard and fast algebraic attacks. Further, we evaluate our parameter for two secondary constructions of Boolean functions. Moreover, A coding-theory approach to the characterization of perfect algebraic immune functions is presented. Via this characterization, infinite families of binary linear complementary dual codes (or LCD codes for short) are obtained from perfect algebraic immune functions. The binary LCD codes presented in this paper have applications in armoring implementations against so-called side-channel attacks (SCA) and fault non-invasive attacks, in addition to their applications in communication and data storage systems.

cs.IT↗

arXiv2020

Full Characterization of Minimal Linear Codes as Cutting Blocking Sets

In this paper, we first study in detail the relationship between minimal linear codes and cutting blocking sets, which were recently introduced by Bonini and Borello, and then completely characterize minimal linear codes as cutting blocking sets. As a direct result, minimal projective codes of dimension $3$ and $t$-fold blocking sets with $t\ge 2$ in projective planes are identical objects. Some bounds on the parameters of minimal codes are derived from this characterization. This confirms a recent conjecture by Alfarano, Borello and Neri in [a geometric characterization of minimal codes and their asymptotic performance, arXiv:1911.11738, 2019] about a lower bound of the minimum distance of a minimal code. Using this new link between minimal codes and blocking sets, we also present new general primary and secondary constructions of minimal linear codes. As a result, infinite families of minimal linear codes not satisfying the Aschikhmin-Barg's condition are obtained. In addition to this, the weight distributions of two subfamilies of the proposed minimal linear codes are established. Open problems are also presented.

cs.IT↗

arXiv2020

A Novel Application of Boolean Functions with High Algebraic Immunity in Minimal Codes

Boolean functions with high algebraic immunity are important cryptographic primitives in some stream ciphers. In this paper, two methodologies for constructing binary minimal codes from sets, Boolean functions and vectorial Boolean functions with high algebraic immunity are proposed. More precisely, a general construction of new minimal codes using minimal codes contained in Reed-Muller codes and sets without nonzero low degree annihilators is presented. The other construction allows us to yield minimal codes from certain subcodes of Reed-Muller codes and vectorial Boolean functions with high algebraic immunity. Via these general constructions, infinite families of minimal binary linear codes of dimension $m$ and length less than or equal to $m(m+1)/2$ are obtained. In addition, a lower bound on the minimum distance of the proposed minimal linear codes is established. Conjectures and open problems are also presented. The results of this paper show that Boolean functions with high algebraic immunity have nice applications in several fields such as symmetric cryptography, coding theory and secret sharing schemes.

cs.IT↗