Nonexistence for extremal Type II $\ZZ_{2k}$-Codes
Tsuyoshi Miezaki
本文证明了当$k=2,3,4,5,6$满足一定条件时，对于所有充分大的$n$，不存在长度为$n$的极值II型$\ZZ_{2k}$ -码.
In this paper, we show that an extremal Type II $\ZZ_{2k}$-code of length $n$ dose not exist for all sufficiently large $n$ when $k=2,3,4,5,6$.
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On the nonexistence of ternary extremal self-dual codes
本文给出了三元极值自对偶码不存在的一个新结果。
The dual codes of the ternary linear codes of the residual designs of biplanes on 56 points are used to prove the nonexistence of quasi-symmetric 2-$(56,12,9)$ and 2-$(57,12,11)$ designs with intersection numbers 0 and 3, and the nonexistence of a 2-$(267,57,12)$ quasi-3 design. The nonexistence of a 2-$(149,37,9)$ quasi-3 design is also proved.
The nonexistence of an additive quaternary [15,5,9]-code
我们证明了不存在加性[15，5，9]_4-码。因此，使加法四元[15，k，9]_4-码存在的最大维数k为k=4.5.
The Golomb--Welch conjecture states that there are no perfect $e$-error-correcting codes in $\mathbb{Z}^n$ for $n \ge 3$ and $e \ge 2$. In this note, we prove the nonexistence of perfect $2$-error-correcting codes for a certain class of $n$, which is expected to be infinite. This result further substantiates the Golomb--Welch conjecture.
On the Existence of Extremal Type II Z2k-Codes
对于长度为8，16和24的码，已知对每一个正整数k都有一个极值Ⅱ型Z2 k-码.本文证明了对每一个正整数k，都存在一个长度为32，40，48，56和64的极值Ⅱ型Z2 k-码.对于长度为72的情况，还证明了对于每个正整数k都存在极值II型Z4 k-码，其中k \ge 2。
It is known that there is no extremal singly even self-dual $[n,n/2,d]$ code with minimal shadow for $(n,d)=(24m+2,4m+4)$, $(24m+4,4m+4)$, $(24m+6,4m+4)$, $(24m+10,4m+4)$ and $(24m+22,4m+6)$. In this paper, we study singly even self-dual codes with minimal shadow having minimum weight $d-2$ for these $(n,d)$. For $n=24m+2$, $24m+4$ and $24m+10$, we show that the weight enumerator of a singly even self-dual $[n,n/2,4m+2]$ code with minimal shadow is uniquely determined and we also show that there is no singly even self-dual $[n,n/2,4m+2]$ code with minimal shadow for $m \ge 155$, $m \ge 156$ an…
Configurations of Extremal Type II Codes
我们证明配置结果的极值II型码，类似于配置结果的Ozeki和第二作者的极值II型格。具体而言，我们表明，对于$n \in \{8, 24, 32, 48, 56, 72, 96\}$的每一个极值长度为$n$的II型码是由其码字的最小重量。其中Ozeki和Kominers使用球谐函数和加权θ函数，我们使用离散谐波多项式和谐波权重计数器。一路上，我们介绍“$t\frac12$ -设计”作为一个离散模拟文科夫的球形设计的相同名称。
On the Existence of Frames of Some Extremal Odd Unimodular Lattices and Self-Dual Zk-Codes
对于维数为n= 12，16，20，32，36，40和44的一些极值（最优）奇幺模格L，我们确定所有正整数k使得L包含k-框架.这一结果给出了长度为12，16，20，32，36，40和44的极值I型Zk-码的存在性，以及对于正整数k，除少数例外，长度为28的近极值I型Zk-码的存在性.
In 1968, Golomb and Welch conjectured that there does not exist perfect Lee code in $\mathbb{Z}^{n}$ with radius $r\ge2$ and dimension $n\ge3$. Besides its own interest in coding theory and discrete geometry, this conjecture is also strongly related to the degree-diameter problems of abelian Cayley graphs. Although there are many papers on this topic, the Golomb-Welch conjecture is far from being solved. In this paper, we prove the nonexistence of linear perfect Lee codes by introducing some new algebraic methods. Using these new methods, we show the nonexistence of linear perfect Lee codes of…
Counterexamples, Constructions, and Nonexistence Results for Optimal Ternary Cyclic Codes
本文通过构造反例和最优码家族，部分解决最优三元循环码的开放问题，并揭示了相关码的约束条件。
We prove new existence and nonexistence results for modular Golomb rulers in this paper. We completely determine which modular Golomb rulers of order $k$ exist, for all $k\leq 11$, and we present a general existence result that holds for all $k \geq 3$. We also derive new nonexistence results for infinite classes of modular Golomb rulers and related structures such as difference packings, optical orthogonal codes, cyclic Steiner systems and relative difference families.
Some Extremal Self-Dual Codes and Unimodular Lattices in Dimension 40
本文研究了40维二元极值单偶自对偶码和40维极值奇幺模格。本文给出了长度为40的极值单偶自对偶码的分类。我们还给出了一个分类的极值奇幺模格在40维的阴影有80个向量的范数2，通过它们的关系，长度为40的极值双偶自对偶码。
The poset metrics that allow binary codes of codimension m to be m-, (m-1)-, or (m-2)-perfect
余维为M的二进制偏序集码（基数为2^{N-M}，其中N是码长）可以纠正最大M个错误。描述了余维数M的码是M-、（M-1）-或（M-2）-完全码的所有可能的偏序集度量。给出了偏序集上不存在完美偏序集码的一般条件，并举例证明了在冠偏序集和不相交链并的情况下，R-完美偏序集码的不存在性.索引术语：完全码，偏序集码
New $s$-extremal extremal unimodular lattices in dimensions $38$, $40$, $42$ and $44$ are constructed from self-dual codes over $\mathbb{F}_5$ by Construction A. In the process of constructing these codes, we obtain a self-dual $[44,22,14]$ code over $\mathbb{F}_5$. In addition, the code implies a $[43,22,13]$ code over $\mathbb{F}_5$. These codes have larger minimum weights than the previously known $[44,22]$ codes and $[43,22]$ codes, respectively.
A relationship between $s$-extremal singly even self-dual $[24k+8,12k+4,4k+2]$ codes and extremal doubly even self-dual $[24k+8,12k+4,4k+4]$ codes with covering radius meeting the Delsarte bound, is established. As an example of the relationship, $s$-extremal singly even self-dual $[56,28,10]$ codes are constructed for the first time. In addition, we show that there is no extremal doubly even self-dual code of length $24k+8$ with covering radius meeting the Delsarte bound for $k \ge 137$. Similarly, we show that there is no extremal doubly even self-dual code of length $24k+16$ with covering r…
On the support designs of extremal binary doubly even self-dual codes
设D是极值二元双偶自对偶[24m，12m，4m+4]码的最小重量的支撑设计。在本说明中，我们考虑当D成为t设计且t \geq为6时的情形。
Extremal Type I $\mathbb{Z}_k$-codes and $k$-frames of odd unimodular lattices
对于维数为$12,16,20,28,32,36,40$和$44$的极值（最优）奇幺模格$L$，我们确定了所有整数$k$使得$L$包含一个$k$ -框架.这个结果产生的存在一个极值I型$\mathbb{Z}_{k}$ -码的长度$12,16,20,32,36,40$和$44$，和一个近极值I型$\mathbb{Z}_k$ -码的长度$28$为正整数$k$只有少数例外。
In this work, we introduce new construction methods for self-dual codes using a Baumert-Hall array. We apply the constructions over the alphabets F_2 and F_4 + uF_4 and combine them with extension theorems and neighboring constructions. As a result, we construct 46 new extremal binary self-dual codes of length 68, 26 new best known Type II codes of length 72 and 8 new extremal Type II codes of length 80 that lead to new 3-(80,16,665) designs. Among the new codes of length 68 are the examples of codes with the rare γ= 5 parameter in W68;2. All these new codes are tabulated in the paper.
An upper bound of the value of t of the support t-designs of extremal binary doubly even self-dual codes
设C是长度为n的极值二元双偶自对偶码，s（C）表示最大整数t，使得C的支撑设计对某个权保持t-设计.本文证明了s（C）\leq 7.
In this work, we introduce the concept of distance between self-dual codes, which generalizes the concept of a neighbor for self-dual codes. Using the k-neighbors, we are able to construct extremal binary self-dual codes of length 68 with new weight enumerators. We construct 143 extremal binary self-dual codes of length 68 with new weight enumerators including 42 codes with gamma=8 in their W_{68,2} and 40 with gamma=9 in their W_{68,2}. These examples are the first in the literature for these gamma values. This completes the theoretical list of possible values for gamma in W_{68,2}.
In this note we report the classification of all symmetric 2-(36,15,6) designs that admit an automorphism of order 2 and their incidence matrices generate an extremal ternary self-dual code. It is shown that up to isomorphism, there exists only one such design, having a full automorphism group of order 24, and the ternary code spanned by its incidence matrix is equivalent to the Pless symmetry code.
Uniformity of extremal graph-codes
研究极值图码及其与密度多项式Hales-Jewett猜想相关问题，揭示极值离散结构的强伪随机行为。
New examples of self-dual near-extremal ternary codes of length 48 derived from 2-(47,23,11) designs
针对长度48的自对偶近极值三元码在$A_{12}$其他取值下的存在性问题，利用带6阶自同构群的对称2-(47,23,11)设计，构造出150个新$A_{12}$取值的该类码，拓展了已知实例范围。
Characterizing extremal digraphs for identifying codes and extremal cases of Bondy's theorem on induced subsets
（di）图$G$的标识码是$G$的顶点的支配子集$C$，使得$G$的所有不同顶点在$C$内具有不同的（内）邻域。本文对所有在任一识别码中只允许其整个顶点集的有限有向图进行了分类。我们也分类所有这样的无限定向图。此外，通过将这一概念与A.邦迪集系统，我们分类的极值情况下，这一定理。
We give a classification of four-circulant singly even self-dual $[60,30,d]$ codes for $d=10$ and $12$. These codes are used to construct extremal singly even self-dual $[60,30,12]$ codes with weight enumerator for which no extremal singly even self-dual code was previously known to exist. From extremal singly even self-dual $[60,30,12]$ codes, we also construct extremal singly even self-dual $[58,29,10]$ codes with weight enumerator for which no extremal singly even self-dual code was previously known to exist. Finally, we give some restriction on the possible weight enumerators of certain si…
On the Residue Codes of Extremal Type II Z4-Codes of Lengths 32 and 40
在本文中，我们确定了长度为32和40的极值II型Z4-码的剩余码的维数。我们证明了每一个长度为32的二进制双偶自对偶码都可以实现为某些极值II型Z4码的剩余码。它还表明，有一个唯一的极值II型Z4码的长度为32，其剩余码具有最小的尺寸6等价。结果，构造了许多新的长度为32和40的极值II型Z4-码。
We give restrictions on the weight enumerators of ternary near-extremal self-dual codes of length divisible by $12$ and quaternary near-extremal Hermitian self-dual codes of length divisible by $6$. We consider the weight enumerators for which there is a ternary near-extremal self-dual code of length $12m$ for $m =3,4,5,6$. Also we consider the weight enumerators for which there is a quaternary near-extremal Hermitian self-dual code of length $6m$ for $m =4,5,6$.
In this note, we demonstrate that every binary doubly even self-dual code of length $40$ can be realized as the residue code of some extremal Type II $\mathbb{Z}_4$-code. As a consequence, it is shown that there are at least $94356$ inequivalent extremal Type II $\mathbb{Z}_4$-codes of length $40$.
Ternary near-extremal self-dual codes of lengths $36$, $48$ and $60$
本文针对长度36、48和60构造了具有新重量计数器的三元近极值自对偶码，填补了该类码此前未知存在的空白。
New extremal binary self-dual codes from F_4 + uF_4-lifts of quadratic double circulant codes over F_4
本文对F_4 + uF_4和F_4应用二次二重循环码和二次加边二重循环码的构造，得到了长度分别为56和64的极值二元自对偶码。利用二进制扩张定理和环扩张形式，得到了7个长度为58的极值自对偶二进制码，24个长度为66的极值自对偶二进制码和29个长度为68的极值自对偶二进制码，它们都具有新的权计数器，更新了文献中所有已知的极值自对偶码的列表.
We construct new $s$-extremal singly even self-dual codes of minimum weights $8,10,12$ and $14$. We also give tables for the currently known results on the existence of $s$-extremal singly even self-dual codes of minimum weights $8,10,12$ and $14$.
Let $C$ be an extremal Type III or IV code and $D_{w}$ be the support design of $C$ for a weight $w$. We introduce the two numbers $δ(C)$ and $s(C)$: $δ(C)$ is the largest integer $t$ such that, for all weights, $D_{w}$ is a $t$-design; $s(C)$ denotes the largest integer $t$ such that there exists a $w$ such that $D_{w}$ is a $t$-design. In the present paper, we consider the possible values of $δ(C)$ and $s(C)$.
We give methods for constructing many self-dual $\mathbb{Z}_m$-codes and Type II $\mathbb{Z}_{2k}$-codes of length $2n$ starting from a given self-dual $\mathbb{Z}_m$-code and Type II $\mathbb{Z}_{2k}$-code of length $2n$, respectively. As an application, we construct extremal Type II $\mathbb{Z}_{2k}$-codes of length $24$ for $k=4,5,\ldots,20$ and extremal Type II $\mathbb{Z}_{2k}$-codes of length $32$ for $k=4,5,\ldots,10$. We also construct new extremal Type II $\mathbb{Z}_4$-codes of lengths $56$ and $64$.
Meta-Fibonacci Sequences, Binary Trees, and Extremal Compact Codes
我们看看一个家庭的元斐波那契序列中出现的研究叶数在最大的水平在某些无穷序列的二叉树，限制组成的整数，和二进制紧凑代码。对于这个家庭的元斐波那契序列和两个家庭的相关序列，我们得到普通的生成函数和递归关系。在这些序列家族中包括在线序列百科全书（OEIS）中的几个众所周知的序列。
For lengths $64$ and $66$, we construct extremal singly even self-dual codes with weight enumerators for which no extremal singly even self-dual codes were previously known to exist. We also construct new $40$ inequivalent extremal doubly even self-dual $[64,32,12]$ codes with covering radius $12$ meeting the Delsarte bound.
A classification of extremal double circulant self-dual codes of lengths up to $88$ is known. We give a classification of extremal double circulant self-dual codes of lengths $90,92,94$ and $96$. We also classify double circulant self-dual codes with parameters $[90,45,14]$ and $[96,48,16]$. In addition, we demonstrate that no double circulant self-dual $[90,45,14]$ code has an extremal self-dual neighbor, and no double circulant self-dual $[96,45,16]$ code has a self-dual neighbor with minimum weight at least $18$.
Weighted Generating Functions for Type II Lattices and Codes
给出了Hamming空间上调和多项式和二元线性码的调和重量计数器的一种新的结构发展，类似于欧几里得空间上调和多项式和欧几里德格的加权θ函数的一种方法.即利用sl 2的有限维表示论，得到了离散齐次多项式空间关于离散调和多项式空间的一个分解定理，并证明了调和权枚举数的一个推广的MacWilliams恒等式.然后，我们提出了几个应用程序的谐波重量枚举，对应于一些使用加权θ函数：t-设计的等价表征，在极端II型码的情况下的阿姆斯-马特森定理，和配置结果的极端II型码的长度为8，24，32，48，56，72和96。
Ternary extremal four-negacirculant self-dual codes
本文研究三元四负循环自对偶码，通过穷举搜索完成了长度40至60的极值码分类。
We introduce an altered version of the four circulant construction over group rings for self-dual codes. We consider this construction over the binary field, the rings F_2 + uF_2 and F_4 + uF_4; using groups of order 3, 7, 9, 13, and 15. Through these constructions and their extensions, we find binary self-dual codes of lengths 32, 40, 56, 64, 68 and 80, all of which are extremal or optimal. In particular, we find five new self-dual codes of parameters [56, 28, 10], twenty-three extremal binary self-dual codes of length 68 with new weight enumerators and fifteen new self-dual codes of paramete…
Locating dominating codes: Bounds and extremal cardinalities
在这项工作中，两种类型的代码，使他们都支配和定位图的顶点进行了研究。这些代码可能是网络中的检测器组，也可能是控制系统的处理器组，它们的一组响应应该确定发生故障的处理器或入侵者。在这里，我们对λ-码和η-码的有关界、极值和实现定理作了较有意义的研究。
On the Classification of Extremal Doubly Even Self-Dual Codes with 2-Transitive Automorphism Group
本文完成了具有2-传递自同构群的极值双偶自对偶码的分类。
In this work, new construction methods for self-dual codes are given. The methods use the short Kharaghani array and a variation of it. These are applicable to any commutative Frobenius ring. We apply the constructions over the ring F_2 + uF_2 and self-dual Type I [64, 32, 12]_2-codes with various weight enumerators obtained as Gray images. By the use of an extension theorem for self-dual codes we were able to construct 27 new extremal binary self-dual codes of length 68. The existence of the extremal binary self-dual codes with these weight enumerators was previously unknown.
In this paper, we construct self-dual codes from a construction that involves both block circulant matrices and block quadratic residue circulant matrices. We provide conditions when this construction can yield self-dual codes. We construct self-dual codes of various lengths over F2 and F2 + uF2. Using extensions, neighbours and sequences of neighbours, we construct many new self-dual codes. In particular, we construct one new self-dual code of length 66 and 51 new self-dual codes of length 68.
On a 5-design related to a putative extremal doubly even self-dual code of length a multiple of 24
根据Assmus和Mattson定理，长度为24 m的极值双偶自对偶码中每个非平凡权的码字构成一个自正交5-设计.本文研究了由具有与上述5-设计相同参数的自正交5-设计构造的码。本文给出了一个自正交5-设计的一些参数，它的存在性等价于一个长度为24 m的极值双偶自对偶码的存在性，其中m=3，…，6.如果$m \in \{1,\ldots,6\}$，$k \in \{m+1,\ldots,5m-1\}$和$(m,k) \ne (6,18)$，则证明了长度为24 m的极值双偶自对偶码是由重量为4k的码字生成的。
New extremal Type II $\mathbb{Z}_4$-codes of length 64 by the doubling method
本文基于倍增法开发了一种构造新极值Type II $\mathbb{Z}_4$-码的方法，并成功构造出三个长度为64、类型为$4^{31}2^2$的新极值Type II $\mathbb{Z}_4$-码，其剩余码为新的最佳已知二元码，且部分码字支撑构成自正交1-设计。
Construction of extremal Type II $\mathbb{Z}_{8}$-codes via doubling method
本文提出倍增方法，从已知极值类型II $\mathbb{Z}_8$-码构造新码，并据此构造出至少十个此前未知的长度为32、类型为$(15,1,1)$的极值类型II $\mathbb{Z}_8$-码，其二元剩余码为最优$[32,15]$二元码。
In this work, connected cubic planar bipartite graphs and related binary self-dual codes are studied. Binary self-dual codes of length 16 are obtained by face-vertex incidence matrices of these graphs. By considering their lifts to the ring R_2 new extremal binary self-dual codes of lengths 64 are constructed as Gray images. More precisely, we construct 15 new codes of length 64. Moreover, 10 new codes of length 66 were obtained by applying a building-up construction to the binary codes. Codes with these weight enumerators are constructed for the first time in the literature. The results are t…
Existence and nonexistence of spherical $5$-designs of minimal type
本文研究极小类型球面5-设计的存在性与性质，证明紧球面5-设计为极小类型的充要条件及相关ETF构造条件，给出其不存在的算术条件，并推导对极球面4-距离5-设计为极小类型的必要条件。
Association schemes related to universally optimal configurations, Kerdock codes and extremal Euclidean line-sets
本文研究了与普遍最优配置、Kerdock码和极值欧几里得直线集相关的关联方案，并探讨了其与Barnes-Wall晶格的联系。
New extremal binary self-dual codes from a modified four circulant construction
本文利用λ-循环矩阵和λ-逆循环矩阵的性质，提出了一种改进的自对偶码的四循环构造法和一种加边构造法。利用$F_2$上的构造，我们得到了长度为64和68的新的二元码.我们还将这些构造应用到环$R_2$上，并考虑$F_2$和$R_1$ -扩张，得到了长度为66和68的新的单偶极值二元自对偶码.更准确地说，我们发现了3个长度为64的新代码，15个长度为66的新代码和22个长度为68的新代码。这些代码都有权重枚举的参数，不知道存在于文献中。
On involutions in extremal self-dual codes and the dual distance of semi self-dual codes
Conway和Pless的一个经典结果是自对偶二元线性码的奇素数阶自同构的固定码的自然投影是自对偶的。在本文中，我们证明了同样的情况下，对合的一些（相当强）的条件下的代码。为了证明这一点，我们引入了一类新的二元码：半自对偶码。一个二元自正交码称为半自对偶码，如果它包含全一向量，并且它的对偶码的余维为2。证明了半自对偶码的对偶距离的上界。作为应用，我们得到如下结果：设C是一个长为24 m的极值自对偶二元线性码，Aut（C）中的s是一个二阶不动点自由自同构.若m是奇数或m=2k且binom{5 k-1}{k-1}是奇数，则C是自由F_2<s>-模.这一结果对这类码的自同构群的结构有很强的影响。
In this work, we generalize the four circulant construction for self-dual codes. By applying the constructions over the alphabets F_2, F_2 + uF_2, F_4+uF_4, we were able to obtain extremal binary self-dual codes of lengths 40, 64 including new extremal binary self-dual codes of length 68. More precisely, 43 new extremal binary self-dual codes of length 68, with rare new parameters have been constructed.
Let $G$ be a finite abelian group, and $r$ be a multiple of its exponent. The generalized Erdős-Ginzburg-Ziv constant $s_r(G)$ is the smallest integer $s$ such that every sequence of length $s$ over $G$ has a zero-sum subsequence of length $r$. We show that $s_{2m}(\mathbb{Z}_2^d) \leq C_m 2^{d/m} + O(1)$ when $d\rightarrow\infty$, and $s_{2m}(\mathbb{Z}_2^d) \geq 2^{d/m} + 2m-1$ when $d=km$. We use results on $s_r(G)$ to prove new bounds for the codegree Turán density of complete $r$-graphs.
Extremal graphs for the identifying code problem
图G的一个标识码是一个支配集C，使得G的每个顶点x与所有其他顶点的区别在于C中距离x至多为1的顶点的集合。找到一个尽可能小的识别码的问题被证明是一个具有挑战性的问题。N. Bertrand证明了如果n个顶点上的至少有一条边的图允许一个标识码，则最小标识码的大小至多为n-1。一些最小标识码为n-1的图类已经被知道，并且很少有公式化的图来分类所有这些图。在本文中，反驳这些命题，我们分类所有的有限图，其中所有的顶点，但其中之一是需要形成一个识别码。我们还分类所有的无限图需要在任何识别码的顶点的整个集合。同时给出了图的顶点数和最大度的新的上界。
We prove that a certain binary linear code associated with the incidence matrix of a quasi-symmetric 2-(37,9,8) design with intersection numbers 1 and 3 must be contained in an extremal doubly even self-dual code of length 40. Using the classification of extremal doubly even self-dual codes of length 40, we show that a quasi-symmetric 2-(37,9,8) design with intersection numbers 1 and 3 does not exist.
Tilings by $(0.5,n)$-Crosses and Perfect Codes
$n$维欧氏空间的十字拼接的存在性问题是众所周知的。一些存在性和不存在性的结果是已知的文献。特别令人感兴趣的是欧几里得空间的十字形镶嵌，其臂长为1，也称为半径为1的李球。这样的镶嵌形成了一个完美的代码。在本文中，交叉与武器的长度一半。这些十字被缩放2以形成离散形状。我们证明了这样一个形状的整数平铺存在当且仅当$n=2^t-1$或$n=3^t-1$，$t>0$。一个强有力的连接，这些平铺的二进制和三进制完美的代码在汉明计划。
For $k \ge 2$ and a positive integer $d_0$, we show that if there exists no quaternary Hermitian linear complementary dual $[n,k,d]$ code with $d \ge d_0$ and Hermitian dual distance greater than or equal to $2$, then there exists no quaternary Hermitian linear complementary dual $[n,k,d]$ code with $d \ge d_0$ and Hermitian dual distance $1$. As a consequence, we generalize a result by Araya, Harada and Saito on the nonexistence of some quaternary Hermitian linear complementary dual codes.
The Automorphism Group of a Self Dual Binary [72,36,16] Code Does Not Contain Z4
在一系列的工作中已经证明了二进制[72，36，16]码的自同构群的阶不超过5。我们得到了所有长度为72的自对偶二进制码的参数化，其自同构为4阶，可以是极值的。我们使用广泛的计算在MAGMA和一台超级计算机上显示，一个极端的二进制代码的长度为72没有一个元素的顺序为4。
Singly-even self-dual codes with minimal shadow
本文研究了具有最小阴影的极值单偶自对偶码。对于特定的参数，我们证明不存在这样的代码。利用Rains \cite{Rains-asymptotic}的一个结果，证明了极值单偶自对偶码的长度是有界的.我们给出明确的界限的情况下，阴影是最小的。
Extremal digraphs for open neighbourhood location-domination and identifying codes
本文研究开邻域定位支配集等于全部顶点的极值有向图，证明其一般结构性质并给出构造方法，同时为已有结果提供新证明并刻画极值有向树。
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