arXiv subjects

Cunsheng Ding

Publications and source records attributed to Cunsheng Ding.

At least 19 recordsLinked to original sources

arXiv2026

Generalized BCH Codes and Twisted Goppa Codes Attaining Their Designed Distances

Determining the true minimum distance of an alternant code remains a notoriously difficult problem in coding theory. In this paper, we study the minimum distances of generalized BCH codes and twisted Goppa codes through their parity-check matrices. We first give a necessary and sufficient condition for an alternant code to attain its designed distance and apply it to generalized BCH codes. As applications, we prove that broad classes of generalized BCH codes have minimum distances equal to their designed distances. These classes provide explicit infinite families rather than isolated examples. We characterize when a twisted Goppa code $\Gamma(L,g,\eta)$ with $\operatorname{deg} g=t$ satisfies $d(\Gamma(L,g,\eta))=t+1$, and derive structured classes and infinite families attaining this distance.

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arXiv2026

Four constructions of self-dual binary cyclic codes with a lower bound on the minimum distances better than the square-root bound

In spite of the intensive study of cyclic codes and the recent construction of an infinite family of self-dual binary cyclic codes whose minimum distances have the square-root bound in IEEE Trans. IT, vol. 71, no. 4, 2025, it is still a 70-year-old open problem whether there is an infinite family of self-dual binary cyclic codes whose minimum distances have a lower bound better than the square-root bound. This paper settles this long-standing open problem in coding theory by presenting infinite families of such self-dual binary cyclic codes. As by-products, several families of cyclic codes with better parameters than those in some references are also constructed in this paper.

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arXiv2026

On the Minimum Distances of Some Families of Goppa Codes and BCH Codes

Goppa codes form an important class of alternant codes with wide applications in algebraic coding theory and code-based cryptography. Determining the true minimum distance of a Goppa code is a difficult problem. In this paper, we provide a necessary and sufficient criterion for a Goppa code to attain its designed distance $\delta=t+1$, where $t$ is the degree of the Goppa polynomial. As applications, we determine the minimum distances of several classes of $q$-ary Goppa codes. In particular, we prove the tightness of the improved lower bound for a class of wild Goppa codes, and extend the family with $G(x)=x^t+A$ from the binary case to arbitrary odd prime powers. We then specialize the criterion to the monomial case $G(x)=x^t$, which is equivalent to primitive BCH codes. This leads to several infinite families of primitive BCH codes with $d=\delta$, including the binary codes $\mathbf{C}_{(2,2^m-1,9,1)}$ and $\mathbf{C}_{(2,2^m-1,15,1)}$, the family $\mathbf{C}_{(p,p^p-1,2p+2,1)}$ with an odd prime $p$ and the family $\mathbf{C}_{(q,q^m-1,r\frac{q^m-1}{q-1}+1,1)}$ with $r\mid q-1$. In particular, we prove that the primitive BCH code $\mathbf{C}_{(q,q^m-1,q^t+1,1)}$ has minimum distance $q^t+1$ under the condition $t\mid m$, improving the previously known condition $pt\mid m$.

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arXiv2026

On the Minimum Distances of Some Families of BCH Codes

BCH codes form an important class of cyclic codes, which have applications in communication and data storage systems. Although the BCH bound provides a lower bound on the minimum distance of BCH codes, determining the true minimum distances of BCH codes is a very challenging problem. In this paper, we settle the minimum distances of a number of infinite families of narrow-sense BCH codes. By explicitly constructing the locator polynomials for minimum weight codewords, we obtain many families of primitive and non-primitive BCH codes with $d=\delta$, where $d$ is the minimum distance of a $q$-ary BCH code of length $n$, designed distance $\delta$, and offset $b$, denoted by $\mathbf{C}_{(q, n, \delta, b)}$. For primitive BCH codes, we obtain infinite families of BCH codes over $\mathbb{F}_3$ and $\mathbb{F}_4$ satisfying $d=\delta$, where $\delta \in \{5,6,7,8\}$. Moreover, we construct several infinite families of $q$-ary BCH codes with $d=\delta$, where $2 \le \delta \le q-1$. For $\delta=q^t+1$, we prove that the BCH code $\mathbf{C}_{(q, q^m-1, q^t+1, 1)}$ has $d=\delta$ for all $m$ satisfying $m \equiv 0 \pmod{pt}$, where $p$ denotes the characteristic of $\mathbb{F}_q$. In the paper by Ding et al., IEEE Trans. Inf. Theory 61(5): 2351-2356, it was conjectured that the minimum distance of $\mathbf{C}_{(q, q^m-1, q^t+1, 1)}$ is always equal to its Bose distance $d_B$. Our result confirms this conjecture for the case $m \equiv 0 \pmod{pt}$. For non-primitive BCH codes, we construct a family of BCH codes $\mathbf{C}_{(q,\frac{q^p-1}{\lambda},p+1,1)}$ with $d=\delta=p+1$, where $p$ is an odd prime, $q=p^e$ with $p \nmid e$ and $\lambda \mid q-1$.

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arXiv2025

Infinitely many families of distance-optimal binary linear codes with respect to the sphere packing bound

R. W. Hamming published the Hamming codes and the sphere packing bound in 1950. In the past 75 years, infinite families of distance-optimal linear codes over finite fields with minimum distance at most 8 with respect to the sphere packing bound have been reported in the literature. However, it is a 75-year-old open problem in coding theory whether there is an infinite family of distance-optimal linear codes over finite fields with arbitrarily large minimum distance with respect to the sphere packing bound. This main objective of this paper is to settle this long-standing open problem in coding theory. As by-products, several infinite families of distance-optimal binary codes with small minimum distances are presented. Two infinite families of binary five-weight codes are reported. Some open problems are also proposed.

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arXiv2025

Some constructions of non-generalized Reed-Solomon MDS Codes

We investigate two classes of extended codes and provide necessary and sufficient conditions for these codes to be non-GRS MDS codes. We also determine the parity check matrices for these codes. Using the connection of MDS codes with arcs in finite projective spaces, we give a new characterization of o-monomials.

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arXiv2024

A Sequence Construction of Cyclic Codes over Finite Fields

Cyclic codes over finite fields are widely implemented in data storage systems, communication systems, and consumer electronics, as they have very efficient encoding and decoding algorithms. They are also important in theory, as they are closely connected to several areas in mathematics. There are a few fundamental ways of constructing all cyclic codes over finite fields, including the generator matrix approach, the generator polynomial approach, the generating idempotent approach, and the $q$-polynomial approach. Another one is a sequence approach, which has been intensively investigated in the past decade. The objective of this paper is to survey the progress in the past decade in this direction.

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arXiv2024

Self-Dual Cyclic Codes with Square-Root-Like Lower Bounds on Their Minimum Distances

Binary self-dual cyclic codes have been studied since the classical work of Sloane and Thompson published in IEEE Trans. Inf. Theory, vol. 29, 1983. Twenty five years later, an infinite family of binary self-dual cyclic codes with lengths $n_i$ and minimum distances $d_i \geq \frac{1}{2} \sqrt{n_i+2}$ was presented in a paper of IEEE Trans. Inf. Theory, vol. 55, 2009. However, no infinite family of Euclidean self-dual binary cyclic codes whose minimum distances have the square-root lower bound and no infinite family of Euclidean self-dual nonbinary cyclic codes whose minimum distances have a lower bound better than the square-root lower bound are known in the literature. In this paper, an infinite family of Euclidean self-dual cyclic codes over the fields ${\bf F}_{2^s}$ with a square-root-like lower bound is constructed. An infinite subfamily of this family consists of self-dual binary cyclic codes with the square-root lower bound. Another infinite subfamily of this family consists of self-dual cyclic codes over the fields ${\bf F}_{2^s}$ with a lower bound better than the square-root bound for $s \geq 2$. Consequently, two breakthroughs in coding theory are made in this paper. An infinite family of self-dual binary cyclic codes with a square-root-like lower bound is also presented in this paper. An infinite family of Hermitian self-dual cyclic codes over the fields ${\bf F}_{2^{2s}}$ with a square-root-like lower bound and an infinite family of Euclidean self-dual linear codes over ${\bf F}_{q}$ with $q \equiv 1 \pmod{4}$ with a square-root-like lower bound are also constructed in this paper.

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arXiv2024

The Support Designs of Several Families of Lifted Linear Codes

A generator matrix of a linear code $\C$ over $\gf(q)$ is also a matrix of the same rank $k$ over any extension field $\gf(q^\ell)$ and generates a linear code of the same length, same dimension and same minimum distance over $\gf(q^\ell)$, denoted by $\C(q|q^\ell)$ and called a lifted code of $\C$. Although $\C$ and their lifted codes $\C(q|q^\ell)$ have the same parameters, they have different weight distributions and different applications. Few results about lifted linear codes are known in the literature. This paper proves some fundamental theory for lifted linear codes, and studies the support $2$-designs of the lifted projective Reed-Muller codes, lifted Hamming codes and lifted Simplex codes. In addition, this paper settles the weight distributions of the lifted Reed-Muller codes of certain orders, and investigates the support $3$-designs of these lifted codes. As a by-product, an infinite family of three-weight projective codes over $\gf(4)$ is obtained.

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arXiv2024

Repeated-Root Cyclic Codes with Optimal Parameters or Best Parameters Known

Cyclic codes are the most studied subclass of linear codes and widely used in data storage and communication systems. Many cyclic codes have optimal parameters or the best parameters known. They are divided into simple-root cyclic codes and repeated-root cyclic codes. Although there are a huge number of references on cyclic codes, few of them are on repeated-root cyclic codes. Hence, repeated-root cyclic codes are rarely studied. There are a few families of distance-optimal repeated-root binary and $p$-ary cyclic codes for odd prime $p$ in the literature. However, it is open whether there exists an infinite family of distance-optimal repeated-root cyclic codes over $\bF_q$ for each even $q \geq 4$. In this paper, three infinite families of distance-optimal repeated-root cyclic codes with minimum distance 3 or 4 are constructed; two other infinite families of repeated-root cyclic codes with minimum distance 3 or 4 are developed; seven infinite families of repeated-root cyclic codes with minimum distance 6 or 8 or 10 are presented; and two infinite families of repeated-root binary cyclic codes with parameters $[2n, k, d \geq (n-1)/\log_2 n]$, where $n=2^m-1$ and $k \geq n$, are constructed. In addition, 27 repeated-root cyclic codes of length up to $254$ over $\bF_q$ for $q \in \{2, 4, 8\}$ with optimal parameters or best parameters known are obtained in this paper. The results of this paper show that repeated-root cyclic codes could be very attractive and are worth of further investigation.

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arXiv2024

Two classes of constacyclic codes with a square-root-like lower bound

Constacyclic codes over finite fields are an important class of linear codes as they contain distance-optimal codes and linear codes with best known parameters. They are interesting in theory and practice, as they have the constacyclic structure. In this paper, an infinite class of $q$-ary negacyclic codes of length $(q^m-1)/2$ and an infinite class of $q$-ary constacyclic codes of length $(q^m-1)/(q-1)$ are constructed and analyzed. As a by-product, two infinite classes of ternary negacyclic self-dual codes with a square-root-like lower bound on their minimum distances are presented.

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arXiv2024

Cyclic and Negacyclic Sum-Rank Codes

Sum-rank codes have known applications in the multishot network coding, the distributed storage and the construction of space-time codes. U. Mart\'ınez-Peñas introduced the cyclic-skew-cyclic sum-rank codes and proposed the BCH bound on the cyclic-skew-cyclic sum-rank codes in his paper published in IEEE Trans. Inf. Theory, vol. 67, no. 8, 2021. Afterwards, many sum-rank BCH codes with lower bounds on their dimensions and minimum sum-rank distances were constructed. Sum-rank Hartmann-Tzeng bound and sum-rank Roos bound on cyclic-skew-cyclic codes were proposed and proved by G. N. Alfarano, F. J. Lobillo, A. Neri, and A. Wachter-Zeh in 2022. In this paper, cyclic, negacyclic and constacyclic sum-rank codes are introduced and a direct construction of cyclic, negacyclic and constacyclic sum-rank codes of the matrix size $m \times m$ from cyclic, negacyclic and constacyclic codes over ${\bf F}_{q^m}$ in the Hamming metric is proposed. The cyclic-skew-cylic sum-rank codes are special cyclic sum-rank codes. In addition, BCH and Hartmann-Tzeng bounds for a type of cyclic sum-rank codes are developed. Specific constructions of cyclic, negacyclic and constacyclic sum-rank codes with known dimensions and controllable minimum sum-rank distances are proposed. Moreover, many distance-optimal binary sum-rank codes and an infinite family of distance-optimal binary cyclic sum-rank codes with minimum sum-rank distance four are constructed. This is the first infinite family of distance-optimal sum-rank codes with minimum sum-rank distance four in the literature.

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arXiv2024

More MDS codes of non-Reed-Solomon type

MDS codes have diverse practical applications in communication systems, data storage, and quantum codes due to their algebraic properties and optimal error-correcting capability. In this paper, we focus on a class of linear codes and establish some sufficient and necessary conditions for them being MDS. Notably, these codes differ from Reed-Solomon codes up to monomial equivalence. Additionally, we also explore the cases in which these codes are almost MDS or near MDS. Applying our main results, we determine the covering radii and deep holes of the dual codes associated with specific Roth-Lempel codes and discover an infinite family of (almost) optimally extendable codes with dimension three.

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arXiv2023

Extended codes and deep holes of MDS codes

For a given linear code $\C$ of length $n$ over $\gf(q)$ and a nonzero vector $\bu$ in $\gf(q)^n$, Sun, Ding and Chen defined an extended linear code $\overline{\C}(\bu)$ of $\C$, which is a generalisation of the classical extended code $\overline{\C}(-\bone)$ of $\C$ and called the second kind of an extended code of $\C$ (see arXiv:2307.04076 and arXiv:2307.08053). They developed some general theory of the extended codes $\overline{\C}(\bu)$ and studied the extended codes $\overline{\C}(\bu)$ of several families of linear codes, including cyclic codes, projective two-weight codes, nonbinary Hamming codes, and a family of reversible MDS cyclic codes. The objective of this paper is to investigate the extended codes $\overline{\C}(\bu)$ of MDS codes $\C$ over finite fields. The main result of this paper is that the extended code $\overline{\C}(\bu)$ of an MDS $[n,k]$ code $\C$ remains MDS if and only if the covering radius $ρ(\mathcal{C}^{\bot})=k$ and the vector $\bu$ is a deep hole of the dual code $\C^\perp$. As applications of this main result, the extended codes of the GRS codes and extended GRS codes are investigated and the covering radii of several families of MDS codes are determined.

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arXiv2023

The Extended Codes of Some Linear Codes

The classical way of extending an $[n, k, d]$ linear code $\C$ is to add an overall parity-check coordinate to each codeword of the linear code $\C$. This extended code, denoted by $\overline{\C}(-\bone)$ and called the standardly extended code of $\C$, is a linear code with parameters $[n+1, k, \bar{d}]$, where $\bar{d}=d$ or $\bar{d}=d+1$. This is one of the two extending techniques for linear codes in the literature. The standardly extended codes of some families of binary linear codes have been studied to some extent. However, not much is known about the standardly extended codes of nonbinary codes. For example, the minimum distances of the standardly extended codes of the nonbinary Hamming codes remain open for over 70 years. The first objective of this paper is to introduce the nonstandardly extended codes of a linear code and develop some general theory for this type of extended linear codes. The second objective is to study this type of extended codes of a number of families of linear codes, including cyclic codes and nonbinary Hamming codes. Four families of distance-optimal or dimension-optimal linear codes are obtained with this extending technique. The parameters of certain extended codes of many families of linear codes are settled in this paper.

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arXiv2023

The extended codes of a family of reversible MDS cyclic codes

A linear code with parameters $[n, k, n-k+1]$ is called a maximum distance separable (MDS for short) code. A linear code with parameters $[n, k, n-k]$ is said to be almost maximum distance separable (AMDS for short). A linear code is said to be near maximum distance separable (NMDS for short) if both the code and its dual are AMDS. MDS codes are very important in both theory and practice. There is a classical construction of a $[q+1, 2u-1, q-2u+3]$ MDS code for each $u$ with $1 \leq u \leq \lfloor\frac{q+1}2\rfloor$, which is a reversible and cyclic code. The objective of this paper is to study the extended codes of this family of MDS codes. Two families of MDS codes and several families of NMDS codes are obtained. The NMDS codes have applications in finite geometry, cryptography and distributed and cloud data storage systems. The weight distributions of some of the extended codes are determined.

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arXiv2023

Four infinite families of ternary cyclic codes with a square-root-like lower bound

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, and the arXiv paper arXiv:2301.06446, the objectives of this paper are the construction and analyses of four infinite families of ternary cyclic codes with length $n=3^m-1$ for odd $m$ and dimension $k \in \{n/2, (n + 2)/2\}$ whose minimum distances have a square-root-like lower bound. Their duals have parameters $[n, k^\perp, d^\perp]$, where $k^\perp \in \{n/2, (n- 2)/2\}$ and $d^\perp$ also has a square-root-like lower bound. These families of codes and their duals contain distance-optimal cyclic codes.

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arXiv2023

Several families of ternary negacyclic codes and their duals

Constacyclic codes contain cyclic codes as a subclass and have nice algebraic structures. Constacyclic codes have theoretical importance, as they are connected to a number of areas of mathematics and outperform cyclic codes in several aspects. Negacyclic codes are a subclass of constacyclic codes and are distance-optimal in many cases. However, compared with the extensive study of cyclic codes, negacyclic codes are much less studied. In this paper, several families of ternary negacyclic codes and their duals are constructed and analysed. These families of negacyclic codes and their duals contain distance-optimal codes and have very good parameters in general.

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