Automorphism groups of cyclic codes

关于二元循环码的自同构群

On automorphism groups of binary cyclic codes

本文研究了二元循环码的完整自同构群，提出了长长度二元循环码的构造方法，以促进完整自同构群的确定。

Automorphism groups with cyclic commutator subgroup and Hamilton cycles

It has been shown that there is a Hamilton cycle in every connected Cayley graph on each group G whose commutator subgroup is cyclic of prime-power order. This paper considers connected, vertex-transitive graphs X of order at least 3 where the automorphism group of X contains a transitive subgroup G whose commutator subgroup is cyclic of prime-power order. We show that of these graphs, only the Petersen graph is not hamiltonian.

$14$-vertex graphs with cyclic automorphism group of order $8$

We describe computational results about undirected graphs having $14$ vertices and automorphism group isomorphic to $\mathbb{Z}/8\mathbb{Z}$, graphs $Γ$ which have less than $2|Aut(Γ)|$ vertices. We give one example of such graphs.

A representation of $\text{Out}\left(F_{n}\right)$ by counting subwords of cyclic words

We generalize the combinatorial approaches of Rapaport and Higgins--Lyndon to the Whitehead algorithm. We show that for every automorphism $φ$ of a free group $F$ and every word $u\in F$ there exists a finite multiset of words $S_{u,φ}$ satisfying the following property: For every cyclic word $w$, the number of times $u$ appears as a subword of $φ\left(w\right)$ depends only on the appearances of words in $S_{u,φ}$ as subwords of $w$. We use this fact to construct a faithful representation of $\text{Out}\left(F_{n}\right)$ on an inverse limit of $\mathbb{Z}$-modules, so that each automorphism …

不可约循环码的排列自同构群

The permutation automorphism groups of irreducible cyclic codes

本文通过有限简单群分类，对不可约循环码的排列自同构群进行了完整分类，证明除少数例外外，非退化不可约循环码均为标准，且为Schmidt-White猜想和Berger-Charpin猜想提供了证据。

An elementary proof of a conjecture on graph-automorphism

In this article, we give an elementary combinatorial proof of a conjecture about the determination of automorphism group of the power graph of finite cyclic groups, proposed by Doostabadi, Erfanian and Jafarzadeh in 2013.

Automorphism Groups and Isometries for Cyclic Orbit Codes

We study orbit codes in the field extension ${\mathbb F}_{q^n}$. First we show that the automorphism group of a cyclic orbit code is contained in the normalizer of the Singer subgroup if the orbit is generated by a subspace that is not contained in a proper subfield of ${\mathbb F}_{q^n}$. We then generalize to orbits under the normalizer of the Singer subgroup. In that situation some exceptional cases arise and some open cases remain. Finally we characterize linear isometries between such codes.

2-Groups that factorise as products of cyclic groups, and regular embeddings of complete bipartite graphs

We classify those 2-groups G which factorise as a product of two disjoint cyclic subgroups A and B, transposed by an automorphism of order 2. The case where G is metacyclic having been dealt with elsewhere, we show that for each e>2 there are exactly three such non-metacyclic groups G with $|A|=|B|=2^e$, and for e=2 there is one. These groups appear in a classification by Berkovich and Janko of 2-groups with one non-metacyclic maximal subgroup; we enumerate these groups, give simpler presentations for them, and determine their automorphism groups.

循环码的置换自同构群 I：分圆结合方案

Permutation automorphism groups of cyclic codes I: cyclotomic association schemes

本文研究循环码的置换自同构群，建立循环码与分圆结合方案的联系，证明 q-仿射群对某些码长不适用，并用算术条件刻画这些码长，其密度为零。

Automorphism groups of rational circulant graphs through the use of Schur rings

The paper concerns the automorphism groups of Cayley graphs over cyclic groups which have a rational spectrum (rational circulant graphs for short). With the aid of the techniques of Schur rings it is shown that the problem is equivalent to consider the automorphism groups of orthogonal group block structures of cyclic groups. Using this observation, the required groups are expressed in terms of generalized wreath products of symmetric groups.

Combinatorial Seifert fibred spaces with transitive cyclic automorphism group

In combinatorial topology we aim to triangulate manifolds such that their topological properties are reflected in the combinatorial structure of their description. Here, we give a combinatorial criterion on when exactly triangulations of 3-manifolds with transitive cyclic symmetry can be generalised to an infinite family of such triangulations with similarly strong combinatorial properties. In particular, we construct triangulations of Seifert fibred spaces with transitive cyclic symmetry where the symmetry preserves the fibres and acts non-trivially on the homology of the spaces. The triangul…

具有给定自同构的线性码的结构研究

On the Structure of the Linear Codes with a Given Automorphism

本文研究有限域上具m阶置换自同构的线性码结构，将其归为广义准循环码，给出准循环码等为自正交、自对偶或线性互补对偶码的充要条件。

Arc-transitive cyclic and dihedral covers of pentavalent symmetric graphs of order twice a prime

A regular cover of a connected graph is called {\em cyclic} or {\em dihedral} if its transformation group is cyclic or dihedral respectively, and {\em arc-transitive} (or {\em symmetric}) if the fibre-preserving automorphism subgroup acts arc-transitively on the regular cover. In this paper, we give a classification of arc-transitive cyclic and dihedral covers of a connected pentavalent symmetric graph of order twice a prime. All those covers are explicitly constructed as Cayley graphs on some groups, and their full automorphism groups are determined.

Digraphs with small automorphism groups that are Cayley on two nonisomorphic groups

Let $Γ=\mathrm{Cay}(G,S)$ be a Cayley digraph on a group $G$ and let $A=\mathrm{Aut}(Γ)$. The Cayley index of $Γ$ is $|A:G|$. It has previously been shown that, if $p$ is a prime, $G$ is a cyclic $p$-group and $A$ contains a noncyclic regular subgroup, then the Cayley index of $Γ$ is superexponential in $p$. We present evidence suggesting that cyclic groups are exceptional in this respect. Specifically, we establish the contrasting result that, if $p$ is an odd prime and $G$ is abelian but not cyclic, and has order a power of $p$ at least $p^3$, then there is a Cayley digraph $Γ$ on $G$ whose …

On abelian-by-cyclic Moufang loops

We study abelian-by-cyclic Moufang loops. We construct all split $3$-divisible abelian-by-cyclic Moufang loops from so-called Moufang permutations on abelian groups $(X,+)$, which are permutations that deviate from an automorphism of $(X,+)$ by an alternating biadditive mapping (satisfying certain properties). More generally, we obtain additional abelian-by-cyclic Moufang loops from so-called construction pairs. As an aside, we show that in the Moufang loops $Q$ obtained from a construction pair on $(X,+)$ the abelian normal subgroup $(X,+)$ induces an abelian congruence of $Q$ if and only if …

具有循环或二面体极大子群的有限2-群及弧传递地图

Finite 2-groups having a cyclic or dihedral maximal subgroup and arc-transitive maps

本文分类了具有循环或二面体极大子群的有限2-群及其自同构群，并据此分类了欧拉示性数不被4整除的弧传递地图。

Automorphism groups of finite posets II

We prove that every finite group $G$ can be realized as the automorphism group of a poset with $4|G|$ points. We also provide bounds for the minimum number of points of a poset with cyclic automorphism group of a given prime power order.

Digraphs from Endomorphisms of Finite Cyclic Groups

We associate each endomorphism of a finite cyclic group with a digraph and study many properties of this digraph, including its adjacent matrix and automorphism group.

Non-standard linear recurring sequence subgroups and automorphisms of irreducible cyclic codes

Let \(\cU\) be the multiplicative group of order~\(n\) in the splitting field \(\bbF_{q^m}\) of \(x^n-1\) over the finite field \(\bbF_q\). Any map of the form \(x\rightarrow cx^t\) with \(c\in \cU\) and \(t=q^i\), \(0\leq i<m\), is \(\bbF_q\)-linear on~\(\bbF_{q^m}\) and fixes \(\cU\) set-wise; maps of this type will be called {\em standard\/}. Occasionally there are other, {\em non-standard\/} \(\bbF_q\)-linear maps on~\(\bbF_{q^m}\) fixing \(\cU\) set-wise, and in that case we say that the pair \((n, q)\) is {\em non-standard\/}. We show that an irreducible cyclic code of length~\(n\) over …

图的雅可比矩阵与图自同构

The Jacobian of a graph and graph automorphisms

本文研究图自同构群在雅可比矩阵对称群中线性表示是否忠实，证明3-边连通图若有非交换半正则自同构群则其雅可比矩阵非循环，并探讨了雅可比矩阵秩的组合意义。

具有几乎交换Terwilliger代数的循环群上的Schur环

Schur rings over cyclic groups having Almost Commutative Terwilliger algebras

分类有限循环群上产生几乎交换Terwilliger代数的轨道Schur环，并给出Schur环的楔积产生几乎交换Terwilliger代数的条件。

Automorphism group of the modified bubble-sort graph

The modified bubble-sort graph of dimension $n$ is the Cayley graph of $S_n$ generated by $n$ cyclically adjacent transpositions. In the present paper, it is shown that the automorphism group of the modified bubble sort graph of dimension $n$ is $S_n \times D_{2n}$, for all $n \ge 5$. Thus, a complete structural description of the automorphism group of the modified bubble-sort graph is obtained. A similar direct product decomposition is seen to hold for arbitrary normal Cayley graphs generated by transposition sets.

Counting the Number of Quasiplatonic Topological Actions of the Cyclic Group on Surfaces

Define $QC(n)$ to be the number of quasiplatonic topological actions of the cyclic group $C_n$ on surfaces of genus at least two. We use formulas of Benim and Wootton to give an explicit formula for $QC(n)$. In addition, we relate the number of quasiplatonic topological actions of $C_n$ to the number of regular dessins d'enfants having $C_n$ as a group of automorphisms.

Graphical Cyclic Permutation Groups

We establish conditions for a permutation group generated by a single permutation of a prime power order to be an automorphism group of a graph or an edge-colored graph. This corrects and generalizes the results of the two papers on cyclic permutation groups published in 1978 and 1981 by S. P. Mohanty, M. R. Sridharan, and S. K. Shukla.

McEliece-type Cryptosystems over Quasi-cyclic Codes

In this thesis, we study algebraic coding theory based McEliece-type cryptosystems over quasi-cyclic codes. The main goal of this thesis is to construct a cryptosystem that resists quantum Fourier sampling making it quantum secure. We propose a new variant of Niederreiter cryptosystem over rate $\frac{m-1}{m}$ quasi-cyclic codes which is secure against quantum Fourier sampling due to indistinguishability of the hidden subgroup. The proof of indistinguishability is achieved due to two constraints over automorphism group; small size and large minimal degree. Apart from this cryptosystem, we also…

实现 $\Z_{2}\times\Z_{2^{n}}$ 为其自同构群的偏序集的最小大小

Minimum Size of a Poset Realizing $\Z_{2}\times\Z_{2^{n}}$ as its Automorphism Group

研究有限群作为有限偏序集自同构群的实现问题，证明对每个 $n\ge 3$，$\beta(\Z_{2}\times\Z_{2^{n}})=2^{\,n+1}+2$。

Wada Dessins associated with Finite Projective Spaces and Frobenius Compatibility

\textit{Dessins d'enfants} (hypermaps) are useful to describe algebraic properties of the Riemann surfaces they are embedded in. In general, it is not easy to describe algebraic properties of the surface of the embedding starting from the combinatorial properties of an embedded dessin. However, this task becomes easier if the dessin has a large automorphism group. In this paper we consider a special type of dessins, so-called \textit{Wada dessins}. Their underlying graph illustrates the incidence structure of finite projective spaces $\PR{m}{n}$. Usually, the automorphism group of these dessin…

A classification of centrally-symmetric and cyclic 12-vertex triangulations of $S^2 \times S^2$

In this paper our main result states that there exist exactly three combinatorially distinct centrally-symmetric 12-vertex-triangulations of the product of two 2-spheres with a cyclic symmetry. We also compute the automorphism groups of the triangulations. These instances suggest that there is a triangulation of $S^2 \times S^2$ with 11 vertices -- the minimum number of vertices required.

实现自同构群为 Z₂ × Z₄ 的最小偏序集

The Minimum Size of a Poset Realizing $\mathbb{Z}_2 \times \mathbb{Z}_4$ as its Automorphism Group

本文证明 β(Z₂ × Z₄) = 14，通过轨道分解和高度函数论证，构造了一个14元偏序集并分析其覆盖关系的刚性。

Quantum automorphism groups of vertex-transitive graphs of order $\leq 11$

We study quantum automorphism groups of vertex-transitive graphs having less than 11 vertices. With one possible exception, these can be obtained from cyclic groups ${\mathbb Z}_n$, symmetric groups $S_n$ and quantum symmetric groups $\mathcal Q_n$, by using various product operations. The exceptional case is that of the Petersen graph, and we present some questions about it.

Nonstandard linear recurring sequence subgroups in finite fields and automorphisms of cyclic codes

Let $q=p^r$ be a prime power, and let $f(x)=x^m-\gs_{m-1}x^{m-1}- >...-\gs_1x-\gs_0$ be an irreducible polynomial over the finite field $\GF(q)$ of size $q$. A zero $ξ$ of $f$ is called {\em nonstandard (of degree $m$) over $\GF(q)$} if the recurrence relation $u_m=\gs_{m-1}u_{m-1} + ... + \gs_1u_1+\gs_0u_0$ with characteristic polynomial $f$ can generate the powers of $ξ$ in a nontrivial way, that is, with $u_0=1$ and $f(u_1)\neq 0$. In 2003, Brison and Nogueira asked for a characterisation of all nonstandard cases in the case $m=2$, and solved this problem for $q$ a prime, and later for $q=p…

Automorphism Groups of nilpotent Lie algebras associated to certain graphs

We consider a family of 2-step nilpotent Lie algebras associated to uniform complete graphs on odd number of vertices. We prove that the symmetry group of such a graph is the holomorph of the additive cyclic group $\Z_n$. Moreover, we prove that the (Lie) automorphism group of the corresponding nilpotent Lie algebra contains the dihedral group of order $2n$ as a subgroup.

On cyclic algebraic-geometry codes

In this paper we initiate the study of cyclic algebraic geometry codes. We give conditions to construct cyclic algebraic geometry codes in the context of algebraic function fields over a finite field by using their group of automorphisms. We prove that cyclic algebraic geometry codes constructed in this way are closely related to cyclic extensions. We also give a detailed study of the monomial equivalence of cyclic algebraic geometry codes constructed with our method in the case of a rational function field.