arXiv subjects

Zhonghua Sun

Publications and source records attributed to Zhonghua Sun.

At least 19 recordsLinked to original sources

arXiv2026

Constructing Good Abelian Codes via Shift Bounds and Genetic Algorithms

This paper investigates the construction of linear codes via abelian codes over finite fields. By exploiting the algebraic structure of multivariate polynomial quotient rings, we derive lower bounds on the minimum distance using a generalized shift bound, which extends the classical van Lint-Wilson bound for cyclic codes. Several infinite families of abelian codes are explicitly constructed, including binary and ternary cases that extend previously known cyclic constructions. To find more abelian codes with good parameters, we apply a genetic algorithm that searches over defining sets represented as binary chromosomes of cyclotomic cosets; the fitness function compares the computed minimum distance against the best known linear code (BKLC) bounds. The search yields multiple record-breaking codes over F_3 and F_4, with improvements over Grassl's tables. Furthermore, the nested structure of these codes enables the application of Construction X, yielding additional linear codes with improved parameters. The results demonstrate that abelian codes, combined with heuristic search, form a viable way for discovering linear codes with unknown parameters.

cs.IT↗

arXiv2026

Generalized Extended Codes with Applications in Entanglement-Assisted Qubit and Qutrit Codes

We prove that any generalized extended code is monomially equivalent to the Hermitian dual of a code which is closely related to a second kind of extended code of $\C^{\perp_{\rm H}}$. Every $[n+1,k+1]_{q^2}$ linear code $\D$ with $d(\D^{\perp_{\rm H}})>1$ is monomially equivalent to the generalized extended code $\C({\bf u},a)$ of an $[n,k]_{q^2}$ linear code $\C$ for a fixed $a\in\F_{q^2}^{*}$ and some ${\bf u}\in\F_{q^2}^{n}$. We then characterize the Hermitian hull and Hermitian dual distance of $\C({\bf u},a)$ in terms of the position of ${\bf u}$ relative to $\C+\C^{\perp_{\rm H}}$ and the interaction between ${\bf u}$ and the minimum weight codewords of $\C^{\perp_{\rm H}}$, respectively. We obtain explicit criteria to independently control the expected Hermitian hull dimension and Hermitian dual distance of $\C({\bf u},a)$. In particular, several conditions for simultaneously increasing the Hermitian hull dimension and the Hermitian dual distance of $\C({\bf u},a)$ are derived. Applying these results to the Hermitian construction for EAQECCs gives us $267$ new EA qubit codes of lengths $n \leq 40$ and $14$ new EA qutrit codes of lengths $n \leq 25$ compared to the best-known codes in Grassl's code tables and the imporvements recorded in very recent works in the literature. Among the new parameter sets, we confirm improvements for $236$ qubit and $8$ qutrit codes.

cs.IT↗

arXiv2025

The Exact Parameters of A Family of BCH Codes

Despite the theoretical and practical significance of BCH codes, the exact minimum distance and dimension remain unknown for many families. This paper establishes the precise minimum distance and dimension of narrow-sense BCH codes $\C_{(q, m, \lambda, \ell_0, \ell_1)}$ over $\gf(q)$ of length $\frac{q^m-1}{\lambda}$ and designed distance $\frac{(q-\lambda \ell_0)q^{m-1-\ell_1}-1}{\lambda}$, where $\lambda\mid (q-1)$, $0\leq \ell_0< \frac{q-1}{\lambda}$, and $0\leq \ell_1\leq m-1$. These results conclusively resolve the three open problems posed by Li et al. (IEEE Trans. Inf. Theory, vol. 63, no. 11, pp. 7219-7236, Nov. 2017) while establishing complementary advances to Ding's seminal framework (IEEE Trans. Inf. Theory, vol. 61, no. 10, pp. 5322-5330, Oct. 2015).

cs.IT↗

arXiv2024

Binary $[n,(n\pm1)/2]$ cyclic codes with good minimum distances from sequences

Recently, binary cyclic codes with parameters $[n,(n\pm1)/2,\geq \sqrt{n}]$ have been a hot topic since their minimum distances have a square-root bound. In this paper, we construct four classes of binary cyclic codes $\mathcal{C}_{\mathcal{S},0}$, $\mathcal{C}_{\mathcal{S},1}$ and $\mathcal{C}_{\mathcal{D},0}$, $\mathcal{C}_{\mathcal{D},1}$ by using two families of sequences, and obtain some codes with parameters $[n,(n\pm1)/2,\geq \sqrt{n}]$. For $m\equiv2\pmod4$, the code $\mathcal{C}_{\mathcal{S},0}$ has parameters $[2^m-1,2^{m-1},\geq2^{\frac{m}{2}}+2]$, and the code $\mathcal{C}_{\mathcal{D},0}$ has parameters $[2^m-1,2^{m-1},\geq2^{\frac{m}{2}}+2]$ if $h=1$ and $[2^m-1,2^{m-1},\geq2^{\frac{m}{2}}]$ if $h=2$.

cs.IT↗

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.

cs.IT↗

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.

cs.IT↗

arXiv2024

A family of linear codes that are either non-GRS MDS codes or NMDS codes

Both maximum distance separable (MDS) codes that are not equivalent to generalized Reed-Solomon (GRS) codes (non-GRS MDS codes) and near MDS (NMDS) codes have nice applications in communication and storage systems. In this paper, we introduce and study a new family of linear codes involving their parameters, weight distributions, and self-orthogonal properties. We prove that such codes are either non-GRS MDS codes or NMDS codes, and hence, they can produce as many of the desired codes as possible. We also completely determine their weight distributions with the help of the solutions to some subset sum problems. A sufficient and necessary condition for such codes to be self-orthogonal is characterized. Based on this condition, we further deduce that there are no self-dual codes in this class of linear codes and explicitly construct two new classes of almost self-dual codes.

cs.IT↗

arXiv2023

A Generalization of the Tang-Ding Binary Cyclic Codes

Cyclic codes are an interesting family of linear codes since they have efficient decoding algorithms and contain optimal codes as subfamilies. Constructing infinite families of cyclic codes with good parameters is important in both theory and practice. Recently, Tang and Ding [IEEE Trans. Inf. Theory, vol. 68, no. 12, pp. 7842--7849, 2022] proposed an infinite family of binary cyclic codes with good parameters. Shi et al. [arXiv:2309.12003v1, 2023] developed the binary Tang-Ding codes to the $4$-ary case. Inspired by these two works, we study $2^s$-ary Tang-Ding codes, where $s\geq 2$. Good lower bounds on the minimum distance of the $2^s$-ary Tang-Ding codes are presented. As a by-product, an infinite family of $2^s$-ary duadic codes with a square-root like lower bound is presented.

cs.IT↗

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.

cs.IT↗

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.

cs.IT↗

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.

cs.IT↗

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.

cs.IT↗

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

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↗

arXiv2020

Femoral Neck Angle Impacts Hip Disorder and Surgical Intervention: A Patient-Specific 3D Printed Analysis

The purpose of this study is to investigate the femoral neck angulation for prediction of the complication associated with dynamic hip screw (DHS) surgery and hip deformity. Three sample patients' MRI images were selected to calculate the femoral neck angles. A total of six femur head geometries were reconstructed and three dimensional (3D) models printed. The calculation of neck angles was done in both computer models and 3D-printed models. Our results showed that 3D-printed models achieved high accuracy and provided the physical measurements, when compared to the computer models could not confirm. Neck angulations related to uncomplicated DHS surgery ranged between 129°-139°, and non-deformity of normal neck angles ranged between 120°-135°. Our study indicated that patient-specific 3D printed femoral head models provide useful information for medical education and assist DHS surgery. Further research based on a large sample size is necessary.

physics.med-ph↗

arXiv2019

A Class of Narrow-Sense BCH Codes

BCH codes are an important class of cyclic codes which have applications in satellite communications, DVDs, disk drives, and two-dimensional bar codes. Although BCH codes have been widely studied, their parameters are known for only a few special classes. Recently, Ding et al. made some new progress in BCH codes. However, we still have very limited knowledge on the dimension of BCH codes, not to mention the weight distribution of BCH codes. In this paper, we generalize the results on BCH codes from several previous papers. The dimension of narrow-sense BCH codes of length $\frac{q^m-1}λ$ with designed distance $2\leq δ\leq \frac{q^{\lceil(m+1)/2 \rceil}-1}λ+1$ is settled, where $λ$ is any factor of $q-1$. The weight distributions of two classes of narrow-sense BCH codes of length $\frac{q^m-1}2$ with designed distance $δ=\frac{(q-1)q^{m-1}-q^{\lfloor(m-1)/2\rfloor}-1}2$ and $δ=\frac{(q-1)q^{m-1}-q^{\lfloor(m+1)/2\rfloor}-1}2$ are determined. The weight distribution of a class of BCH codes of length $\frac{q^m-1}{q-1}$ is determined. In particular, a subclass of this class of BCH codes is optimal with respect to the Griesmer bound. Some optimal linear codes obtained from this class of BCH codes are characterized.

cs.IT↗

arXiv2018

Reversible Codes and Its Application to Reversible DNA Codes over $F_{4^k}$

Coterm polynomials are introduced by Oztas et al. [a novel approach for constructing reversible codes and applications to DNA codes over the ring $F_2[u]/(u^{2k}-1)$, Finite Fields and Their Applications 46 (2017).pp. 217-234.], which generate reversible codes. In this paper, we generalize the coterm polynomials and construct some reversible codes which are optimal codes by using $m$-quasi-reciprocal polynomials. Moreover, we give a map from DNA $k$-bases to the elements of $F_{4^k}$, and construct reversible DNA codes over $F_{4^k}$ by DNA-$m$-quasi-reciprocal polynomials.

cs.IT↗