arXiv subjects

Cunsheng Ding

Publications and source records attributed to Cunsheng Ding.

At least 37 records · Page 2Linked to original sources

arXiv2023

Five infinite families of binary cyclic codes and their related codes with good parameters

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 contain distance-optimal codes. The other two families of binary cyclic codes are composed of binary duadic codes with a square-root-like lower bound on their minimum distances. As a by-product, two infinite families of self-dual binary codes with a square-root-like lower bound on their minimum distances are obtained.

cs.IT↗

arXiv2022

Permutation trinomials over $\mathbb{F}_{2^m}$: a corrected version

Permutation polynomials are an interesting subject of mathematics and have applications in other areas of mathematics and engineering. In this paper, we determine all permutation trinomials over $\mathbb{F}_{2^m}$ in Zieve's paper. We prove a conjecture proposed by Gupta and Sharma and obtain some new permutation trinomials over $\mathbb{F}_{2^m}$. Finally, we show that some classes of permutation trinomials with parameters are QM equivalent to some known permutation trinomials.

math.CO↗

arXiv2022

Two Classes of Constacyclic Codes with Variable Parameters

Constacyclic codes over finite fields are a family of linear codes and contain cyclic codes as a subclass. Constacyclic codes are related to many areas of mathematics and outperform cyclic codes in several aspects. Hence, constacyclic codes are of theoretical importance. On the other hand, constacyclic codes are important in practice, as they have rich algebraic structures and may have efficient decoding algorithms. In this paper, two classes of constacyclic codes are constructed using a general construction of constacyclic codes with cyclic codes. The first class of constacyclic codes is motivated by the punctured Dilix cyclic codes and the second class is motivated by the punctured generalised Reed-Muller codes. The two classes of constacyclic codes contain optimal linear codes. The parameters of the two classes of constacyclic codes are analysed and some open problems are presented in this paper.

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

Several Families of Irreducible Constacyclic and Cyclic Codes

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 prime such that $\ord_n(q)=(n-1)/2$. The results in this paper complement earlier works on irreducible constacyclic and cyclic codes over finite fields.

cs.IT↗

arXiv2022

Two families of negacyclic BCH codes

Negacyclic BCH codes are a subclass of neagcyclic codes and are the best linear codes in many cases. However, there have been very few results on negacyclic BCH codes. Let $q$ be an odd prime power and $m$ be a positive integer. The objective of this paper is to study negacyclic BCH codes with length $\frac{q^m-1}{2}$ and $\frac{q^m+1}{2}$ over the finite field $\mathbf(q)$ and analyse their parameters. The negacyclic BCH codes presented in this paper have good parameters in general, and contain many optimal linear codes. For certain $q$ and $m$, compared with cyclic codes with the same dimension and length, the negacyclic BCH codes presented in this paper have a larger minimum distance in some cases.

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↗

arXiv2021

The minimum linear locality of linear codes

Locally recoverable codes (LRCs) were proposed for the recovery of data in distributed and cloud storage systems about nine years ago. A lot of progress on the study of LRCs has been made by now. However, there is a lack of general theory on the minimum linear locality of linear codes. In addition, the minimum linear locality of many known families of linear codes is not studied in the literature. Motivated by these two facts, this paper develops some general theory about the minimum linear locality of linear codes, and investigates the minimum linear locality of a number of families of linear codes, such as $q$-ary Hamming codes, $q$-ary Simplex codes, generalized Reed-Muller codes, ovoid codes, maximum arc codes, the extended hyperoval codes, and near MDS codes. Many classes of both distance-optimal and dimension-optimal LRCs are presented in this paper. The minimum linear locality of many families of linear codes are settled with the general theory developed in this paper.

cs.IT↗

arXiv2021

Some punctured codes of several families of binary linear codes

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 families of binary linear codes with few weights were obtained in the literature. This paper studies some punctured codes of these binary codes. Several families of binary linear codes with few weights and new parameters are obtained in this paper. Several families of distance-optimal binary linear codes with new parameters are also produced in this paper.

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

The construction and weight distributions of all projective binary linear codes

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.

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

The Subfield Codes of $[q+1, 2, q]$ MDS Codes

Recently, subfield codes of geometric codes over large finite fields $\gf(q)$ with dimension $3$ and $4$ were studied and distance-optimal subfield codes over $\gf(p)$ were obtained, where $q=p^m$. The key idea for obtaining very good subfield codes over small fields is to choose very good linear codes over an extension field with small dimension. This paper first presents a general construction of $[q+1, 2, q]$ MDS codes over $\gf(q)$, and then studies the subfield codes over $\gf(p)$ of some of the $[q+1, 2,q]$ MDS codes over $\gf(q)$. Two families of dimension-optimal codes over $\gf(p)$ are obtained, and several families of nearly optimal codes over $\gf(p)$ are produced. Several open problems are also proposed in this paper.

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

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↗

arXiv2020

Optimal Binary Linear Codes from Maximal Arcs

The binary Hamming codes with parameters $[2^m-1, 2^m-1-m, 3]$ are perfect. Their extended codes have parameters $[2^m, 2^m-1-m, 4]$ and are distance-optimal. The first objective of this paper is to construct a class of binary linear codes with parameters $[2^{m+s}+2^s-2^m,2^{m+s}+2^s-2^m-2m-2,4]$, which have better information rates than the class of extended binary Hamming codes, and are also distance-optimal. The second objective is to construct a class of distance-optimal binary codes with parameters $[2^m+2, 2^m-2m, 6]$. Both classes of binary linear codes have new parameters.

cs.IT↗