arXiv subjects

Dongchun Han

Publications and source records attributed to Dongchun Han.

10 recordsLinked to original sources

arXiv2026

On zero-sum polytopes: reciprocity, rigidity, and cyclic sieving

Let $G$ be a finite abelian group of order $n$, and let $\mathsf M(G,m)$ denote the set of zero-sum sequences over $G$ of length $m$. We introduce the zero-sum polytope $\mathcal P_G$, a rational polytope of dimension $n-1$, whose lattice points encode zero-sum sequences: \[ |\mathsf M(G,m)|=|m\mathcal P_G\cap \mathbb Z^n|. \] This naturally realizes the enumeration of zero-sum sequences as a problem in rational Ehrhart theory, which leads to a combinatorial reciprocity theorem identifying the negative evaluations of the corresponding counting quasipolynomial with zero-sum sequences of full support. Our main results establish a face-stratified rigidity for zero-sum polytopes: whenever two such polytopes have equal total lattice point counts at specific dilations, the dimension-wise open-face strata are equinumerous. Moreover, we study the natural $\operatorname{Aut}(G)$-action on $\mathcal P_G$, derive equivariant generating functions and reciprocity formulas, and obtain cyclic sieving phenomena for natural cyclic actions.

math.CO↗

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.

math.CO↗

arXiv2023

New constructions of NMDS self-dual codes

Near maximum distance separable (NMDS) codes are important in finite geometry and coding theory. Self-dual codes are closely related to combinatorics, lattice theory, and have important application in cryptography. In this paper, we construct a class of $q$-ary linear codes and prove that they are either MDS or NMDS which depends on certain zero-sum condition. In the NMDS case, we provide an effective approach to construct NMDS self-dual codes which largely extend known parameters of such codes. In particular, we proved that for square $q$, almost $q/8$ NMDS self-dual $q$-ary codes can be constructed.

math.CO↗

arXiv2022

Near-MDS Codes from Maximal Arcs in PG$(2,q)$

The singleton defect of an $[n,k,d]$ linear code ${\cal C}$ is defined as $s({\cal C})=n-k+1-d$. Codes with $S({\cal C})=0$ are called maximum distance separable (MDS) codes, and codes with $S(\cal C)=S(\cal C ^{\bot})=1$ are called near maximum distance separable (NMDS) codes. Both MDS codes and NMDS codes have good representations in finite projective geometry. MDS codes over $F_q$ with length $n$ and $n$-arcs in PG$(k-1,q)$ are equivalent objects. When $k=3$, NMDS codes of length $n$ are equivalent to $(n,3)$-arcs in PG$(2,q)$. In this paper, we deal with the NMDS codes with dimension 3. By adding some suitable projective points in maximal arcs of PG$(2,q)$, we can obtain two classes of $(q+5,3)$-arcs (or equivalently $[q+5,3,q+2]$ NMDS codes) for any prime power $q$. We also determine the exact weight distribution and the locality of such NMDS codes and their duals. It turns out that the resultant NMDS codes and their duals are both distance-optimal and dimension-optimal locally recoverable codes.

cs.IT↗

arXiv2021

A reciprocity on finite abelian groups involving zero-sum sequences

In this paper, we present a reciprocity on finite abelian groups involving zero-sum sequences. Let $G$ and $H$ be finite abelian groups with $(|G|,|H|)=1$. For any positive integer $m$, let $\mathsf M(G,m)$ denote the set of all zero-sum sequences over $G$ of length $m$. We have the following reciprocity $$|\mathsf M(G,|H|)|=|\mathsf M(H,|G|)|.$$ Moreover, we provide a combinatorial interpretation of the above reciprocity using ideas from rational Catalan combinatorics. We also present and explain some other symmetric relationships on finite abelian groups with methods from invariant theory. Among others, we partially answer a question proposed by Panyushev in a generalized version.

math.CO↗

arXiv2021

On $*$-clean group rings over finite fields

A ring $R$ is called clean if every element of $R$ is the sum of a unit and an idempotent. Motivated by a question proposed by Lam on the cleanness of von Neumann Algebras, Vaš introduced a more natural concept of cleanness for $*$-rings, called the $*$-cleanness. More precisely, a $*$-ring $R$ is called a $*$-clean ring if every element of $R$ is the sum of a unit and a projection ($*$-invariant idempotent). Let $\mathbb F$ be a finite field and $G$ a finite abelian group. In this paper, we introduce two classes of involutions on group rings of the form $\mathbb FG$ and characterize the $*$-cleanness of these group rings in each case. When $*$ is taken as the classical involution, we also characterize the $*$-cleanness of $\mathbb F_qG$ in terms of LCD abelian codes and self-orthogonal abelian codes in $\mathbb F_qG$.

math.RA↗

arXiv2019

On an open problem about a class of optimal ternary cyclic codes

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems and communication systems as they have efficient encoding and decoding algorithms. In this paper, we settle an open problem about a class of optimal ternary cyclic codes which was proposed by Ding and Helleseth. Let $C_{(1,e)}$ be a cyclic code of length $3^m-1$ over GF(3) with two nonzeros $α$ and $α^e$, where $α$ is a generator of $GF(3^m)^*$ and e is a given integer. It is shown that $C_{(1,e)}$ is optimal with parameters $[3^m-1,3^m-1-2m,4]$ if one of the following conditions is met. 1) $m\equiv0(\mathrm{mod}~ 4)$, $m\geq 4$, and $e=3^\frac{m}{2}+5$. 2) $m\equiv2(\mathrm{mod}~ 4)$, $m\geq 6$, and $e=3^\frac{m+2}{2}+5$.

math.CO↗

arXiv2018

On generalized Erdős-Ginzburg-Ziv constants of $C_n^r$

Let $G$ be an additive finite abelian group with exponent $\exp(G)=m$. For any positive integer $k$, the $k$-th generalized Erdős-Ginzburg-Ziv constant $\mathsf s_{km}(G)$ is defined as the smallest positive integer $t$ such that every sequence $S$ in $G$ of length at least $t$ has a zero-sum subsequence of length $km$. It is easy to see that $\mathsf s_{kn}(C_n^r)\ge(k+r)n-r$ where $n,r\in\mathbb N$. Kubertin conjectured that the equality holds for any $k\ge r$. In this paper, we mainly prove the following results: (1) For every positive integer $k\ge 6$, we have $$\mathsf s_{kn}(C_n^3)=(k+3)n+O(\frac{n}{\ln n}).$$ (2) For every positive integer $k\ge 18$, we have $$\mathsf s_{kn}(C_n^4)=(k+4)n+O(\frac{n}{\ln n}).$$ (3) For $n\in \mathbb N$, assume that the largest prime power divisor of $n$ is $p^a$ for some $a\in\mathbb N$. For any fixed $r\ge 5$, if $p^t\ge r$ for some $t\in\mathbb N$, then for any $k\in\mathbb N$ we have $$\mathsf s_{kp^tn}(C_n^r)\le(kp^t+r)n+c_r\frac{n}{\ln n},$$ where $c_r$ is a constant depends on $r$. Note that the main terms in our results are consistent with the conjectural values proposed by Kubertin.

math.CO↗

arXiv2018

Erdős-Ginzburg-Ziv theorem and Noether number for $C_m\ltimes_φ C_{mn}$

Let $G$ be a multiplicative finite group and $S=a_1\cdot\ldots\cdot a_k$ a sequence over $G$. We call $S$ a product-one sequence if $1=\prod_{i=1}^ka_{τ(i)}$ holds for some permutation $τ$ of $\{1,\ldots,k\}$. The small Davenport constant $\mathsf d(G)$ is the maximal length of a product-one free sequence over $G$. For a subset $L\subset \mathbb N$, let $\mathsf s_L(G)$ denote the smallest $l\in\mathbb N_0\cup\{\infty\}$ such that every sequence $S$ over $G$ of length $|S|\ge l$ has a product-one subsequence $T$ of length $|T|\in L$. Denote $\mathsf e(G)=\max\{\text{ord}(g): g\in G\}$. Some classical product-one (zero-sum) invariants including $\mathsf D(G):=\mathsf s_{\mathbb N}(G)$ (when $G$ is abelian), $\mathsf E(G):=\mathsf s_{\{|G|\}}(G)$, $\mathsf s(G):=\mathsf s_{\{\mathsf e(G)\}}(G)$, $η(G):=\mathsf s_{[1,\mathsf e(G)]}(G)$ and $\mathsf s_{d\mathbb N}(G)$ ($d\in\mathbb N$) have received a lot of studies. The Noether number $β(G)$ which is closely related to zero-sum theory is defined to be the maximal degree bound for the generators of the algebra of polynomial invariants. Let $G\cong C_m\ltimes_φ C_{mn}$, in this paper, we prove that $$\mathsf E(G)=\mathsf d(G)+|G|=m^2n+m+mn-2$$ and $β(G)=\mathsf d(G)+1=m+mn-1$. We also prove that $\mathsf s_{mn\mathbb N}(G)=m+2mn-2$ and provide the upper bounds of $η(G)$, $\mathsf s(G)$. Moreover, if $G$ is a non-cyclic nilpotent group and $p$ is the smallest prime divisor of $|G|$, we prove that $β(G)\le \frac{|G|}{p}+p-1$ except if $p=2$ and $G$ is a dicyclic group, in which case $β(G)=\frac{1}{2}|G|+2$.

math.CO↗

arXiv2018

On Davenport constant of finite abelian groups

$G$ be an additive finite abelian group. The Davenport constant $\mathsf D(G)$ is the smallest integer $t$ such that every sequence (multiset) $S$ over $G$ of length $|S|\ge t$ has a non-empty zero-sum subsequence. Recently, B. Girard proved that for every fixed integer $r > 1$ the Davenport constant $\mathsf D(C_n^r)$ is asymptotic to $rn$ when $n$ tends to infinity. In this paper, for every fixed positive integer $r$, we prove that $$\mathsf D(C_n^r)=rn+O(\frac{n}{\ln n}).$$ This is an explicit version of the above result of B. Girard. Furthermore, we can get better estimates of the error term for some $n$ of special types. Finally, we get an asymptotic result for some finite abelian groups of special types. Our proof combines a classical argument in the zero-sum theory together with some basic tools and results from analytic number theory.

math.CO↗