arXiv subjects

Chunming Tang

Publications and source records attributed to Chunming Tang.

At least 19 recordsLinked to original sources

arXiv2026

A globally and superlinearly convergent QO-free method for nonlinear optimization on Riemannian manifolds

The quadratic optimization-free (QO-free) method is a class of powerful and effective algorithms for solving nonlinearly constrained optimization problems in Euclidean spaces. The aim of the present work is to extend this method to solve optimization problems on manifolds with additional equality and inequality constraints. We first present a specific algorithm in the manifold setting. At each iteration, three linear systems sharing a common linear operator are solved to determine the master search direction. In addition, a higher-order correction direction is obtained by solving a reduced linear least squares subproblem to circumvent the Maratos effect which is assumed not to arise in existing related literature. A Riemannian arc search is then performed within the tangent space of the current iterate to generate the new iterate. Under appropriate assumptions, we establish the global and strong convergence of the proposed method. Moreover, we prove that the unit step size will eventually be accepted by the arc search, upon which the superlinear convergence of the algorithm is established. Finally, numerical results demonstrate that the proposed method is very competitive compared with other existing approaches.

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arXiv2026

When Do Subset Sums in Finite Abelian Groups Support $2$-Designs?

Subset sums over finite abelian groups lie at the intersection of additive combinatorics, design theory, and coding theory. Let $G$ be a finite abelian group, and let $\cB_k^x$ be the family of $k$-subsets of $G$ whose elements sum to $x\in G$. This paper studies when the incidence structure $(G,\cB_k^x)$ is a block design. The elementary abelian $p$-group case was settled by Falcone and Pavone. Pavone (\emph{Des. Codes Cryptogr.} 91 (2023), 2585--2603) further asked whether, for an arbitrary finite abelian group $G$, the zero-sum incidence structure $(G,\cB_k^0)$ can be a nontrivial $2$-design only when $G$ is an elementary abelian $p$-group. We settle this open question in the stronger form that, for every $x\in G$, $(G,\cB_k^x)$ can be a nontrivial $2$-design only if $G$ is an elementary abelian $p$-group. The proof develops a character-theoretic approach to subset-sum designs, using character sums over the blocks to constrain the structure of the character group $\widehat G$. The approach also yields a complete characterization of subset-sum $1$-designs and general arithmetic restrictions on subset-sum designs over arbitrary finite abelian groups, extending the corresponding results previously known for finite abelian $p$-groups.

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arXiv2026

Beyond Polynomials: Optimal Locally Recoverable Codes from Good Rational Functions

Locally recoverable codes (LRCs) have emerged as fundamental objects in modern coding theory, primarily due to their pivotal role in distributed and cloud storage systems. A major breakthrough in their construction was achieved by Tamo and Barg, who introduced the notion of \emph{good polynomials} as a key structural ingredient. In this article, we propose a natural generalization of this paradigm by introducing the concept of \emph{good rational functions}. Building upon this extension, we develop a unified and flexible framework for constructing optimal LRCs. To quantify the quality of a rational function, we embed the problem into the rich context of algebraic function field theory and Galois theory. This perspective allows us to extend the Galois-theoretic framework originally developed by Micheli for good polynomials. In particular, we derive structural and quantitative results on the number of totally split rational places associated with rational functions. Furthermore, we construct explicit families of good rational functions that outperform all good polynomials of the same degree. As a consequence, we obtain infinite families of optimal LRCs with improved parameters compared to those arising from the classical Tamo-Barg construction. These results highlight the intrinsic strength of our approach.

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arXiv2026

Rank Distribution and Dynamics of Gram Matrices from Binary m-Sequences with Applications to LCD Codes

The Gram matrix is a classical object formed from the pairwise inner products of a collection of vectors, with fundamental roles in functional analysis, statistics, combinatorics, and coding theory. In the realm of sequence design, maximum-length sequences (m-sequences) are among the most fundamental classes of sequences, traditionally characterized by their span, decimation, shift-and-add, balance, run, and ideal autocorrelation properties. In this paper, we bridge the two foundational concepts by uncovering novel structural features of m-sequences through the lens of a family of Gram matrices. Specifically, for each $1 \le t \le 2^n - 1$, we extract $n$ consecutive subsequences of length $t$ from an m-sequence of period $2^n - 1$, construct their corresponding $n \times n$ Gram matrix, and investigate its rank, denoted by $r_n(t)$. Utilizing semilinear representation of Galois groups and B\'ezoutian of polynomials, we derive an explicit formula for $r_n(t)$ for all $t$, thereby establishing the complete rank distribution of these Gram matrices. Notably, we prove that full rank is attained for approximately half of the admissible values of $t$. We further uncover the intricate dynamics of $r_n(t)$: rank-deficient states are strictly unstable (i.e., $r_n(t) < n$ implies $r_n(t+1) \ne r_n(t)$), whereas the full-rank state exhibits strong persistence, remaining at $n$ over a nontrivial interval of consecutive values of $t$. Altogether, our results fully characterize both the global rank distribution and the local dynamics of rank function, as invariant of m-sequences. As an application, our findings completely determine the hull distribution of the family of punctured cyclic simplex codes.

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arXiv2026

Improved Penalty Function Approaches for Optimization Problems with General Orthogonality

In this paper, we consider a class of generalized orthogonal optimization constraint problems (GOOCP) over $\mathbb{R}^{n \times p}$, where the variable $X$ is restricted within the intersection of a certain subspace $\mathcal{F}$ and satisfies the quadratic constraint $\{X \in \mathbb{R}^{n \times p}: X^{\top} \phi(X) = I_p\}$. Such constraints generalize a wide range of structured matrix manifolds, such as the Stiefel manifold, the symplectic Stiefel manifold, the indefinite Stiefel manifold, the third-order tensor Stiefel manifold, etc. We show that the feasible region of GOOCP is a closed embedded submanifold of $\mathbb{R}^{n \times p}$ and characterize the necessary geometric materials for the existing Riemannian optimization frameworks. Based on the constraint dissolving approach for Riemannian optimization problems, we propose the constraint dissolving penalty function (GOCDF) for the constrained optimization problem GOOCP with easy-to-compute formulations. We further establish the equivalence between GOCDF and GOOCP in the aspects of first-order and second-order stationary points. We also analyze the computational complexity of applying first-order methods to minimize GOOCP, which could be significantly lower than those of first-order Riemannian optimization methods. Numerical experiments demonstrate that solving GOOCP through applying unconstrained optimization methods to minimize constraint dissolving function demonstrates superior efficiency to existing Riemannian optimization methods.

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arXiv2026

The Asymmetric Hamming Bidistance and Distributions over Binary Asymmetric Channels

The binary asymmetric channel is a model for practical communication systems where the error probabilities for symbol transitions $0\rightarrow 1$ and $1\rightarrow0$ differ substantially. In this paper, we introduce the notion of asymmetric Hamming bidistance (AHB) and its two-dimensional distribution, which separately captures directional discrepancies between codewords. This finer characterization enables a more discriminative analysis of decoding the error probabilities for maximum-likelihood decoding (MLD), particularly when conventional measures, such as weight distributions and existing discrepancy-based bounds, fail to distinguish code performance. Building on this concept, we derive a new upper bound on the average error probability for binary codes under MLD and show that, in general, it is incomparable with the two existing bounds derived by Cotardo and Ravagnani (IEEE Trans. Inf. Theory, 68 (5), 2022). To demonstrate its applicability, we compute the complete AHB distributions for several families of codes, including two-weight and three-weight projective codes (with the zero codeword removed) via strongly regular graphs and 3-class association schemes, as well as nonlinear codes constructed from symmetric balanced incomplete block designs (SBIBDs).

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arXiv2026

Non-RS type cyclic MDS codes over finite fields via cyclotomic field reduction

Cyclic maximum distance separable (MDS for short) codes are a special subclass of linear codes and have received a lot of attention, as these codes have very important applications in many areas including quantum codes, designs and finite geometry. However, the existing construction methods for cyclic MDS codes are mainly focused on strict restrictions on certain parameters or are relatively complex in their construction approaches. In this paper, we investigate this approach further via norm reduction in cyclotomic fields. By converting the verification of the MDS property over a finite field into checking non-zero minors in characteristic zero, we propose a construction method of cyclic MDS codes over finite fields via cyclotomic field reduction. Based on this method, we obtain several cyclic MDS codes over finite fields and many non-RS type cyclic MDS codes are produced. Compared with the existing construction methods, our method is relatively simpler. Moreover, the results of this paper show that the parameters of the obtained non-RS cyclic MDS codes are flexible.

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arXiv2025

Multi-Twisted Generalized Reed-Solomon Codes: Structure, Properties, and Constructions

Maximum distance separable (in short, MDS), near MDS (in short, NMDS), and self-orthogonal codes play a pivotal role in algebraic coding theory, particularly in applications such as quantum communications and secret sharing scheme. Recently, the construction of non-generalized Reed-Solomon (in short, non-GRS) codes has emerged as a significant research frontier. This paper presents a systematic investigation into a generalized class of $(\mathcal{L}, \mathcal{P})$-twisted generalized Reed-Solomon (TGRS) codes characterized by $\ell$ twists, extending the structures previously introduced by Beelen et al. and Hu et al.. We first derive the explicit parity-check matrices for these codes by analyzing the properties of symmetric polynomials. Based on this algebraic framework, we establish necessary and sufficient conditions for the self-orthogonality of the proposed codes, generalizing several recent results. Leveraging these self-orthogonal structures, we construct new families of LCD MDS codes that offer greater flexibility in code length compared to existing literature. Furthermore, we provide a characterization of the NMDS property for these codes, offering a partial solution to the open problem concerning general $(\mathcal{L}, \mathcal{P})$-TGRS codes posed by Hu et al. (2025). Finally, we rigorously prove that these codes are of non-GRS type when $2k > n$, providing an improvement over previous bounds. Theoretical constructions are validated through numerical examples.

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arXiv2025

Capacity-Achieving Codes for Noisy Insertion Channels

DNA storage has emerged as a promising solution for large-scale and long-term data preservation. Among various error types, insertions are the most frequent errors occurring in DNA sequences, where the inserted symbol is often identical or complementary to the original, and in practical implementations, noise can further cause the inserted symbol to mutate into a random one, which creates significant challenges to reliable data recovery. In this paper, we investigate a new noisy insertion channel, where infinitely many insertions of symbols complement or identical to the original ones and up to one insertion of random symbol may occur. We determine the coding capacity of the noisy channel and construct asymptotically optimal error-correcting codes achieving the coding capacity.

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arXiv2025

A Riemannian conjugate subgradient method for nonconvex and nonsmooth optimization on manifolds

Conjugate gradient (CG) methods are widely acknowledged as efficient for minimizing continuously differentiable functions in Euclidean spaces. In recent years, various CG methods have been extended to Riemannian manifold optimization, but existing Riemannian CG methods are confined to smooth objective functions and cannot handle nonsmooth ones. This paper proposes a Riemannian conjugate subgradient method for a class of nonconvex, nonsmooth optimization problems on manifolds. Specifically, we first select a Riemannian subgradient from the convex hull of two directionally active subgradients. The search direction is then defined as a convex combination of the negative of this subgradient and the previous search direction transported to the current tangent space. Additionally, a Riemannian line search with an interval reduction procedure is integrated to generate an appropriate step size, ensuring the objective function values form a monotonically nonincreasing sequence. We establish the global convergence of the algorithm under mild assumptions. Numerical experiments on three classes of Riemannian optimization problems show that the proposed method takes significantly less computational time than related existing methods. To our knowledge, this is the first CG-type method developed for Riemannian nonsmooth optimization.

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arXiv2025

A Generic Construction of $q$-ary Near-MDS Codes Supporting 2-Designs with Lengths Beyond $q+1$

A linear code with parameters $[n, k, n - k + 1]$ is called maximum distance separable (MDS), and one with parameters $[n, k, n - k]$ is called almost MDS (AMDS). A code is near-MDS (NMDS) if both it and its dual are AMDS. NMDS codes supporting combinatorial $t$-designs have attracted growing interest, yet constructing such codes remains highly challenging. In 2020, Ding and Tang initiated the study of NMDS codes supporting 2-designs by constructing the first infinite family, followed by several other constructions for $t > 2$, all with length at most $q + 1$. Although NMDS codes can, in principle, exceed this length, known examples supporting 2-designs and having length greater than $q + 1$ are extremely rare and limited to a few sporadic binary and ternary cases. In this paper, we present the first \emph{generic construction} of $q$-ary NMDS codes supporting 2-designs with lengths \emph{exceeding $q + 1$}. Our method leverages new connections between elliptic curve codes, finite abelian groups, subset sums, and combinatorial designs, resulting in an infinite family of such codes along with their weight distributions.

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arXiv2025

Combinatorial $t$-Designs from Finite Abelian Groups and Their Applications to Elliptic Curve Codes

In this paper, we establish the conditions for some finite abelian groups and the family all the $k$-sets in each of them summing up to an element $x$ to form $t$-designs. We fully characterize the sufficient and necessary conditions for the incidence structures to form $1$-designs in finite abelian $p$-groups, generalizing existing results on vector spaces over finite fields. For finite abelian groups of exponent $pq$, we also propose sufficient and necessary conditions for the incidence structures to form a $1$-designs. Furthermore, some interesting observations of the general case when the group is cyclic or non-cyclic are presented and the relations between $(t-1)$-designs and $t$-designs from subset sums are established. As an application, we demonstrate the correspondence between $t$-designs from the minimum-weight codewords in elliptic curve codes and subset-sum designs in their groups of rational points. By such a correspondence, elliptic curve codes supporting designs can be simply derived from subset sums in finite abelian groups that supporting designs.

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arXiv2025

Enhancing Blockchain Cross Chain Interoperability: A Comprehensive Survey

Blockchain technology, introduced in 2008, has revolutionized data storage and transfer across sectors such as finance, healthcare, intelligent transportation, and the metaverse. However, the proliferation of blockchain systems has led to discrepancies in architectures, consensus mechanisms, and data standards, creating data and value silos that hinder the development of an integrated multi chain ecosystem. Blockchain interoperability (a.k.a cross chain interoperability) has thus emerged as a solution to enable seamless data and asset exchange across disparate blockchains. In this survey, we systematically analyze over 150 high impact sources from academic journals, digital libraries, and grey literature to provide an in depth examination of blockchain interoperability. By exploring the existing methods, technologies, and architectures, we offer a classification of interoperability approaches including Atomic Swaps, Sidechains, Light Clients, and so on, which represent the most comprehensive overview to date. Furthermore, we investigate the convergence of academic research with industry practices, underscoring the importance of collaborative efforts in advancing blockchain innovation. Finally, we identify key strategic insights, challenges, and future research trajectories in this field. Our findings aim to support researchers, policymakers, and industry leaders in understanding and harnessing the transformative potential of blockchain interoperability to address current challenges and drive forward a cohesive multi-chain ecosystem.

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arXiv2024

Deformed Intersections of Half-spaces

This paper is devoted to the classification problems concerning extended deformations of convex polyhedra and real hyperplane arrangements in the following senses: combinatorial equivalence of face posets, normal equivalence on normal fans of convex polyhedra, and sign equivalence on half-spaces. The extended deformations of convex polyhedra arise from parallel translations of given half-spaces and hyperplanes, whose normal vectors give rise to the so-called ``derived arrangement'' proposed by Rota as well as Crapo in different forms. We show that two extended deformations of convex polyhedra are normally (combinatorially, as a consequence) equivalent if they are parameterized by the same open face of the derived arrangement. Note that these extended deformations are based on parallel translations of the given hyperplanes. It allows us to study three deformations of real hyperplane arrangements: parallel translations, conings, and elementary lifts, whose configuration spaces are parameterized by open faces of the derived arrangement. Consequently, it gives a characterization of the normal, combinatorial, and sign equivalences of those three deformations via the derived arrangement. Additionally, the relationships among these three equivalence relations are discussed, and several new descriptions of real derived arrangements associated with faces and sign vectors of real hyperplane arrangements are provided.

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arXiv2024

A restricted memory quasi-Newton bundle method for nonsmooth optimization on Riemannian manifolds

In this paper, a restricted memory quasi-Newton bundle method for minimizing a locally Lipschitz continuous function over a Riemannian manifold is proposed. The curvature information of the objective function is approximated by applying a Riemannian version of the quasi-Newton updating formulas. A Riemannian subgradient aggregation technique is proposed and used to significantly reduce the computations in the quadratic programming subproblem when calculating the candidate descent direction. Moreover, a Riemannian line-search procedure is proposed to generate the stepsizes, and the process is finitely terminated under the assumption of a newly proposed Riemannian semismoothness. Global convergence of the proposed method is established: if the serious iteration steps are finite, then the last serious iterate is stationary; otherwise, every accumulation point of the serious iteration sequence is stationary. In addition, a modified algorithm with limited-memory quasi-Newton updates is presented to further reduce the computational cost. Finally, numerical experiments demonstrate that (i) the quasi-Newton updates accelerate the convergence of the bundle method, (ii) the aggregation technique significantly reduces the computational cost for solving the quadratic programming subproblem, and (iii) the proposed methods outperform the compared state-of-the-art Riemannian optimization methods for locally Lipschitz continuous functions.

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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.

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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.

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arXiv2023

A descent method for nonsmooth multiobjective optimization problems on Riemannian manifolds

In this paper, a descent method for nonsmooth multiobjective optimization problems on complete Riemannian manifolds is proposed. The objective functions are only assumed to be locally Lipschitz continuous instead of convexity used in existing methods. A necessary condition for Pareto optimality in Euclidean space is generalized to the Riemannian setting. At every iteration, an acceptable descent direction is obtained by constructing a convex hull of some Riemannian $\varepsilon$-subgradients. And then a Riemannian Armijo-type line search is executed to produce the next iterate. The convergence result is established in the sense that a point satisfying the necessary condition for Pareto optimality can be generated by the algorithm in a finite number of iterations. Finally, some preliminary numerical results are reported, which show that the proposed method is efficient.

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