Discrete Mathematics最新文献

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Remarks on proper conflict-free degree-choosability of graphs with prescribed degeneracy 给定简并度图的适当无冲突度可选性

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-06-01 Epub Date: 2026-01-28 DOI: 10.1016/j.disc.2026.115003

Masaki Kashima , Riste Škrekovski , Rongxing Xu

A proper coloring ϕ of G is called a proper conflict-free coloring of G if for every non-isolated vertex v of G, there is a color c such that

| ϕ^{- 1} (c) \cap N_{G} (v) | = 1

. As an analogy of degree-choosability of graphs, we introduced the notion of proper conflict-free

(degree + k)

-choosability of graphs. For a non-negative integer k, a graph G is proper conflict-free

(degree + k)

-choosable if for any list assignment L of G with

| L (v) | \geq d_{G} (v) + k

for every vertex

v \in V (G)

, G admits a proper conflict-free coloring ϕ such that

ϕ (v) \in L (v)

for every vertex

v \in V (G)

. In this note, we first remark if a graph G is d-degenerate, then G is proper conflict-free

(degree + d + 1)

-choosable. Furthermore, when

d = 1

, we can reduce the number of colors by showing that every tree is proper conflict-free

(degree + 1)

-choosable. This motivates us to state a question.

G的适当着色φ称为G的适当无冲突着色，如果对于G的每一个非孤立顶点v，存在一个颜色c使得| φ - 1(c)∩NG(v)|=1。作为图的可选择度的类比，我们引入了图的适当无冲突（度+k）可选择性的概念。对于非负整数k，图G是适当的无冲突（度+k）可选的，如果对于G的任意列表赋值L，对于每个顶点v∈v (G)，具有|L(v)|≥dG(v)+k，则G承认一个适当的无冲突着色φ，使得对于每个顶点v∈v (G), φ (v)∈L(v)。在本文中，我们首先注意到，如果一个图G是d-简并的，那么G是适当的无冲突（度+d+1）可选的。此外，当d=1时，我们可以通过显示每棵树都是适当的无冲突（度+1）可选来减少颜色的数量。这促使我们提出一个问题。

引用次数: 0

Perfect proper edge colorings of regular bipartite graphs with rainbow C4-s 具有彩虹C4-s的正则二部图的完备真边着色

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-06-01 Epub Date: 2026-01-16 DOI: 10.1016/j.disc.2026.115005

András Gyárfás , Gábor N. Sárközy , Adam Zsolt Wagner

<div><div>We call a proper edge coloring of a bipartite graph <math><mi>G</mi><mo>=</mo><mo>[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>]</mo></math> with <math><mo>|</mo><mi>X</mi><mo>|</mo><mo>=</mo><mo>|</mo><mi>Y</mi><mo>|</mo><mo>=</mo><mi>n</mi></math> a B-coloring if every 4-cycle of G is colored with four different colors. Denote by <math><msub><mrow><mi>q</mi></mrow><mrow><mi>B</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math> the smallest number of colors needed for a B-coloring of graph G. The question whether <math><msub><mrow><mi>q</mi></mrow><mrow><mi>B</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>n</mi></math> implies <math><mo>|</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mo>=</mo><mi>o</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math> is from Burr, Erdős, Graham and Sós. A positive answer to this question would imply a positive answer to the famous <math><mo>(</mo><mn>7</mn><mo>,</mo><mn>4</mn><mo>)</mo></math>-conjecture of Brown, Erdős and Sós. Here we look at an interesting test case of this question. We call a B-coloring of a d-regular bipartite graph <math><mi>G</mi><mo>=</mo><mo>[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>]</mo></math> with <math><mo>|</mo><mi>X</mi><mo>|</mo><mo>=</mo><mo>|</mo><mi>Y</mi><mo>|</mo><mo>=</mo><mi>n</mi></math> perfect if each color class forms a perfect matching in G (i.e. has n edges).</div><div>Let <math><mi>f</mi><mo>(</mo><mi>d</mi><mo>)</mo></math> be the minimal n such that there exists a perfect B-coloring of some d-regular bipartite graph <math><mi>G</mi><mo>=</mo><mo>[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>]</mo></math> with <math><mo>|</mo><mi>X</mi><mo>|</mo><mo>=</mo><mo>|</mo><mi>Y</mi><mo>|</mo><mo>=</mo><mi>n</mi></math>. A test case of the question above is whether <math><mi>f</mi><mo>(</mo><mi>d</mi><mo>)</mo></math> is super-linear. We prove the affirmative answer for shifted colorings: defined by the matchings <math><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi></mrow></msub></math><math><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mrow><mo>{</mo><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>k</mi><mo>+</mo><mi>i</mi></mrow></msub><mo>)</mo><mspace></mspace><mo>|</mo><mspace></mspace><mn>0</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>i</mi><mo>∈</mo><mi>D</mi><mo>}</mo></mrow><mo>,</mo></math> where <math><mi>D</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>i</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>i</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>}</mo><

对于具有|X|=|Y|=n的二部图G=[X,Y]，如果G的每4个循环都有4种不同的颜色，我们称其为b -染色。用qB(G)表示图G的b -着色所需的最小颜色数。qB(G)=n是否意味着|E(G)|=o（n2）的问题来自Burr， Erdős， Graham和Sós。对这个问题的肯定回答意味着对Brown， Erdős和Sós著名的（7,4）猜想的肯定回答。下面我们来看这个问题的一个有趣的测试案例。我们称d正则二部图G=[X，Y]且|X|=|Y|=n的b -着色，如果每个颜色类在G中形成一个完美匹配（即有n条边）。设f(d)为最小n，使得某d正则二部图G=[X，Y]具有|X|=|Y|=n的完美b -着色。上述问题的一个测试用例是f(d)是否超线性。我们证明了移位着色的肯定答案：由MiMi={(xk,yk+i)|0≤k≤n−1，i∈D}定义，其中D={i1，…，id}∧[0,n−1]，k+i是模n计算的。利用szemer的诱导匹配引证，我们将这个结果推广到更一般的着色类，我们称之为平移不变着色。一般地，我们证明了对于奇数d 2d+1≤f(d)，对于偶d 2d - 1≤f(d)，并且等式只对某些对称设计的关联图成立。从Behrend构造我们得到f(d)≤declog (d)我们利用小双翼和对称设计的性质证明了f(3)=7,f(4)=8,f(5)=12,12≤f(6)≤14,f(7)=16。

引用次数: 0

Density of 2-critical signed graphs 2临界符号图的密度

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-06-01 Epub Date: 2026-01-14 DOI: 10.1016/j.disc.2026.114998

Reza Naserasr , Weiqiang Yu

A balanced k-coloring of a signed graph

(G, σ)

which has no negative loop is to partition its vertices into k sets each of which induces a balanced subgraph, that is a subgraph with no negative cycle. The notion, through basic graph operation, captures the classic proper coloring of graphs as special case.

Having observed the importance of balanced 2-coloring, in this work we study structural conditions which permit a signed graph to admit a 2-coloring. More precisely, slightly modifying the notion of color-critical, we say a signed graph is k-critical if it admits no balanced k-coloring but every proper subgraph of it admits such a coloring.

We show that if

(G, σ)

is a 2-critical signed graph whose underlying graph is not

K_{5}

or an odd cycle, then

| E (G) | \geq \frac{21 | V (G) | - 2 d + 1}{10}

where d is the maximum number of vertex disjoint digons in

(G, σ)

. As a corollary we conclude that, except for the signed graph

(K_{5}, -)

, any signed simple graph with maximum average degree at most 4.2 admits a balanced 2-coloring.

无负环的有符号图（G,σ）的平衡k-着色是将其顶点划分为k个集，每个集引出一个平衡子图，即无负环的子图。这个概念通过基本的图运算，将经典的图的适当着色作为特例。在观察到平衡2-着色的重要性之后，本文研究了允许一个符号图允许2-着色的结构条件。更准确地说，稍微修改一下色临界的概念，我们说一个有符号图是k临界的，如果它不允许平衡的k染色，但它的每个固有子图都允许这样的染色。证明了如果（G,σ）是一个底图不是K5或奇环的2临界符号图，则|E(G)|≥21|V(G)|−2d+110，其中d为（G,σ）中顶点不相交双子的最大个数。作为推论，我们得出，除了有符号图（K5,−）外，任何最大平均度不超过4.2的有符号简单图都是平衡的2-着色。

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引用次数: 0

Structure and linear-Pollyanna for some square-free graphs 一些无平方图的结构和线性波利安娜

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-06-01 Epub Date: 2026-01-08 DOI: 10.1016/j.disc.2026.114979

Ran Chen, Baogang Xu

We use

P_{t}

and

C_{t}

to denote a path and a cycle on t vertices, respectively. A bull is a graph consisting of a triangle with two disjoint pendant edges, a hammer is a graph obtained by identifying an endvertex of a

P_{3}

with a vertex of a triangle. A class

F

is χ-bounded if there is a function f such that

χ (G) \leq f (ω (G))

for all induced subgraphs G of a graph in

F

. A class

C

of graphs is Pollyanna (resp. linear-Pollyanna) if

C \cap F

is polynomially (resp. linearly) χ-bounded for every χ-bounded class

F

of graphs. Chudnovsky et al. [6] showed that both the classes of bull-free graphs and hammer-free graphs are Pollyannas. Let G be a connected graph with no clique cutsets and no universal vertices. In this paper, we show that G is

(C_{4}

, hammer)-free if and only if it has girth at least 5, and G is

(C_{4}

, bull)-free if and only if it is a clique blowup of some graph of girth at least 5. As a consequence, we show that both the classes of

(C_{4}

, bull)-free graphs and

(C_{4}

, hammer)-free graphs are linear-Pollyannas. We also show that the class of (bull, diamond)-free graphs is linear-Pollyanna.

我们用Pt和Ct分别表示t个顶点上的路径和循环。牛是由一个三角形的两个不相交的垂边组成的图，锤是由P3的一个端点与一个三角形的一个顶点确定而成的图。如果存在一个函数F，使得F中一个图的所有诱导子图G的χ(G)≤F (ω(G))，则该类F是χ-有界的。如果C∩F是多项式的(p。对于每一个有χ有界的图类F。Chudnovsky et al.[6]表明无牛图和无锤图都是盲目乐观的。设G是一个连通图，没有团切集，也没有全称顶点。本文证明了G是（C4，锤子）自由的当且仅当它的周长至少为5，并且G是（C4，公牛）自由的当且仅当它是某个周长至少为5的图的团团爆破。因此，我们证明了（C4，牛）自由图和（C4，锤）自由图都是线性的盲目乐观。我们还证明了一类（牛，菱形）无图是线性的。

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引用次数: 0

Deletable edges in 3-connected graphs and their applications 3连通图中的可删除边及其应用

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-06-01 Epub Date: 2026-01-23 DOI: 10.1016/j.disc.2026.114994

S.R. Kingan

<div><div>Let G and H be simple 3-connected graphs such that G has an H-minor. An edge e in G is called H-deletable if <math><mi>G</mi><mo>﹨</mo><mi>e</mi></math> is 3-connected and has an H-minor. The main result in this paper establishes that, if G has no H-deletable edges, then there exists a sequence of simple 3-connected graphs <math><msub><mrow><mi>G</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow></msub></math> with no H-deletable edges such that <math><msub><mrow><mi>G</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>≅</mo><mi>H</mi></math>, <math><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><mi>G</mi></math>, and for <math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>k</mi></math> one of three possibilities holds: <math><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>/</mo><mi>f</mi></math>; <math><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>/</mo><mi>f</mi><mo>﹨</mo><mi>e</mi></math> where e and f are incident to a degree 3 vertex in <math><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub></math>; or <math><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>−</mo><mi>w</mi></math> where w is a degree 3 vertex in <math><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub></math>. We give several applications including a graph-theoretic proof of the matroid theory result known as the Strong Splitter Theorem, a short new proof of Dirac's characterization of 3-connected graphs with no minor isomorphic to the prism graph, and an extension of a result by Halin that bounds the number of edges in a minimally 3-connected graph. Halin proved that if G is a minimally 3-connected graph on <math><mi>n</mi><mo>≥</mo><mn>8</mn></math> vertices, then <math><mo>|</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mo>≤</mo><mn>3</mn><mi>n</mi><mo>−</mo><mn>9</mn></math> and equality holds if and only if <math><mi>G</mi><mo>≅</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn><mo>,</mo><mi>n</mi><mo>−</mo><mn>3</mn></mrow></msub></math>. We give a different proof of Halin's result and extend it by identifying the minimally 3-connected infinite family of graphs with <math><mo>|</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mo>=</mo><mn>3</mn><mi>n</mi><mo>−</mo><mn>10</mn></math>. Finally, we extend the main theorem to mat

设G和H是简单的3连通图，其中G有一个H小调。G中的边e被称为h可删除边，如果Ge是3连通的并且有一个h小调。本文的主要结果证明，如果G没有H-可删除边，则存在一个无H-可删除边的简单3连通图序列G0，…，Gk，使得G0 = H, Gk=G，且对于1≤i≤k，有三种可能之一成立：Gi−1=Gi/f；Gi−1=Gi/fe，其中e和f入射到Gi中的3次顶点；或者Gi−1=Gi−w，其中w是Gi中的一个3度顶点。我们给出了几个应用，包括矩阵理论结果强分裂定理的图论证明，Dirac关于无小同构棱镜图的3连通图刻划的一个简短的新证明，以及Halin关于最小3连通图边数限定的一个结果的推广。Halin证明了如果G是n≥8个顶点上的最小3连通图，则|E(G)|≤3n−9，且等式成立当且仅当G = K3,n−3。我们给出了Halin结果的另一种证明，并通过识别|E(G)|=3n−10的最小3连通无限族图来扩展它。最后，将主要定理推广到拟阵中。

引用次数: 0

On non-traceable 3-connected planar cubic graphs of minimum order 最小阶不可追溯三连通平面三次图

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-06-01 Epub Date: 2026-01-12 DOI: 10.1016/j.disc.2026.114983

Gholamreza Abrishami , Freydoon Rahbarnia , Nico Van Cleemput

In 1980, Zamfirescu presented a non-traceable (i.e. without a hamiltonian path) 3-connected planar cubic graph with 88 vertices, which is still the smallest known one of its kind. In this note we present several new examples with 88 vertices. Moreover, we present a non-traceable cyclically 4-connected planar cubic graph with 168 vertices.

1980年，Zamfirescu提出了一个88个顶点的不可追踪（即没有哈密顿路径）三连通平面三次图，这是目前已知的最小的三连通平面三次图。在这篇文章中，我们提供了几个新的88个顶点的例子。此外，我们给出了一个不可追踪的有168个顶点的环四连通平面三次图。

引用次数: 0

Solutions to open problems on the exponential augmented Zagreb index 指数增广萨格勒布指数开放性问题的解法

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-06-01 Epub Date: 2026-01-06 DOI: 10.1016/j.disc.2025.114967

Kinkar Chandra Das , Da-yeon Huh , Sourav Mondal

The exponential augmented Zagreb (EAZ) index is a graph-theoretical descriptor that correlates strongly with the physico-chemical properties of molecules. Introduced by Rada in 2019, it is defined for a simple graph ϒ as

E A Z (ϒ) = \sum_{v_{i} v_{j} \in E (ϒ)} e^{{(\frac{d_{i} d_{j}}{d_{i} + d_{j} - 2})}^{3}},

where

E (ϒ)

denotes the edge set and

d_{i}

is the degree of vertex

v_{i}

. This work is motivated by some open problems concerning the well-known augmented Zagreb index (AZ). In particular, the maximization of AZ for a given graph order and a specified number of pendant vertices was posed as an open problem in Chen et al. (2022) [7]. We completely resolve this problem for the exponential version, EAZ. In recent work Xu et al. (2025) [38], two related questions were raised: whether the maximal graphs for AZ and EAZ coincide, and if not, how they differ. We provide complete answers to these questions with respect to the chromatic number and the number of pendant vertices. We explore the maximal graph for EAZ in terms of chromatic number and graph order, and show that this differs substantially from the corresponding extremal graph for AZ. Further results include a characterization of the maximal graphs for EAZ when vertex connectivity and edge connectivity are prescribed together with the graph order. In addition, we prove that

E A Z (ϒ)

increases upon adding an edge to ϒ, a crucial result for understanding the extremal properties of EAZ. Finally, the potential usefulness of this discrete invariant in chemical graph theory is demonstrated.

指数增强萨格勒布指数（EAZ）是一种与分子的物理化学性质密切相关的图理论描述符。由Rada于2019年引入，它被定义为一个简单的图表γ asEAZ(y)=∑vivj∈E(y) E(didjdi+dj−2)3，其中E(y)表示边缘集，di表示顶点vi的度。这项工作的灵感来自于一些关于著名的增强萨格勒布指数（AZ）的开放问题。特别是，在Chen等人（2022）[7]中，给定图阶和指定数量的垂顶点的AZ最大化是一个开放问题。我们完全解决了这个问题的指数版本，EAZ。在最近的工作Xu et al. (2025) b[38]中，提出了两个相关的问题：AZ和EAZ的最大图是否重合，如果不重合，它们是如何不同的。我们提供了关于色数和垂顶点数的完整答案。我们从色数和图阶的角度探讨了EAZ的极大图，并表明这与AZ的相应极值图有很大的不同。进一步的结果包括了当顶点连通性和边连通性与图阶一起规定时EAZ的极大图的表征。此外，我们证明了在给y增加一个边缘后，y的值会增加，这是理解y的极端属性的一个关键结果。最后，证明了该离散不变量在化学图论中的潜在用途。

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引用次数: 0

Improved bounds for proper rainbow saturation 改进了彩虹饱和度的边界

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-06-01 Epub Date: 2026-01-28 DOI: 10.1016/j.disc.2026.115001

Andrew Lane , Natasha Morrison

Given a graph H, we say that a graph G is properly rainbow H-saturated if: (1) There is a proper edge colouring of G containing no rainbow copy of H; (2) For every

e \notin E (G)

, every proper edge colouring of

G + e

contains a rainbow copy of H. The proper rainbow saturation number

sa t^{⁎} (n, H)

is the minimum number of edges in a properly rainbow H-saturated graph. In this paper we use connections to the classical saturation and semi-saturation numbers to provide new upper bounds on

sa t^{⁎} (n, H)

for general cliques, cycles, and complete bipartite graphs. We also provide a necessary and sufficient condition for a graph to have constant proper rainbow saturation number.

给定一个图H，我们说图G是适当的彩虹H饱和的，如果：(1)有一个G的适当的边缘着色，不包含H的彩虹副本；(2)对于每一个e (G)， G+e的每一个适当的边着色都包含一个H的彩虹副本，其中适当的彩虹饱和数sat _ （n，H）是一个适当的彩虹H饱和图的最小边数。在本文中，我们利用经典饱和和半饱和数的连接，为一般团、环和完全二部图提供了新的上界（n，H）。并给出了图具有常固有彩虹饱和数的充分必要条件。

引用次数: 0

The Kneser chromatic function distinguishes trees 克奈瑟色函数区分树

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-06-01 Epub Date: 2026-01-20 DOI: 10.1016/j.disc.2026.115009

Yusaku Nishimura

R.P. Stanley defined an invariant for graphs called the chromatic symmetric function and conjectured it is a complete invariant for trees. Miezaki et al. generalized the chromatic symmetric function and defined the Kneser chromatic functions denoted by

{X_{K_{N, k}}}_{k \in N}

, and rephrase Stanley's conjecture that

X_{K_{N, 1}}

is a complete invariant for trees. This paper shows

X_{K_{N, 2}}

is a complete invariant for trees.

R.P. Stanley定义了一个图的不变量，称为色对称函数，并推测它是树的完全不变量。miiezaki等人推广了色对称函数，定义了Kneser色函数{XKN，k}k∈N，并重新表述了Stanley关于XKN，1是树的完全不变量的猜想。本文证明XKN，2是树的完全不变量。

引用次数: 0

Quasi-cyclic codes of index 2 索引2的拟循环码

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-06-01 Epub Date: 2026-01-19 DOI: 10.1016/j.disc.2026.115004

Kanat Abdukhalikov , Askar S. Dzhumadil'daev , San Ling

We study quasi-cyclic codes of index 2 over finite fields. We give a classification of such codes. Their duals with respect to the Euclidean, symplectic and Hermitian inner products are investigated. We describe self-orthogonal and dual-containing codes. Lower bounds for minimum distances of quasi-cyclic codes are given. A quasi-cyclic code of index 2 is generated by at most two elements. We describe conditions when such a code (or its dual) is generated by one element.

研究有限域上指标2的拟循环码。我们对这类代码进行了分类。研究了它们对欧几里得内积、辛内积和厄米内积的对偶。我们描述了自正交和双包含码。给出了拟循环码最小距离的下界。索引为2的拟循环码由最多两个元素生成。我们将描述由一个元素生成这种代码（或其对偶）的条件。

引用次数: 0

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