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Presenting sporadic simple groups as an automorphism groups of designs and graphs is an exciting field in finite group theory. In this paper, with two different methods, we present some new resolvable simple 3-designs with Higman-Sims sporadic simple group HS as the full automorphism group. Also, we classify all block-transitive self-orthogonal designs on 176 points with even block size that admit sporadic simple group HS as an automorphism group. Furthermore, with these methods we construct some new resolvable 3-designs on 36, 40, 120 and 176 points. [ABSTRACT FROM AUTHOR]

Consider G to be a finite group and p to be a prime divisor of the order | G | in the group G. The main aim of this paper is to prove that the outcome in a recent paper of A. Laradji is true in the case of a p-constrained group. We observe that the generalization of the concept of Navarro's vertex for an irreducible character in a p-constrained group G is generally undefined. We illustrate this with a suitable example. Let ϕ ∈ I r r (G) have a positive height, and let there be an anchor group A ϕ . We prove that if the normalizer N G (A ϕ) is p-constrained, then O p ´ (N G (A ϕ)) ≠ { 1 G } , where O p ´ (N G (A ϕ)) is the maximal normal p ´ subgroup of N G (A ϕ) . We use character theoretic methods. In particular, Clifford theory is the main tool used to accomplish the results. [ABSTRACT FROM AUTHOR]

In this paper, we first give a relationship between generalized algebraic geometry codes (GAG codes) and algebraic geometry codes (AG codes). More precisely, we show that a GAG code is contained (up to isomorphism) in a suitable AG code. Next we recall the concept of an N1N2-automorphism group, a subgroup of the automorphism group of a GAG code. With the use of the relation we obtained between these two classes of codes, we show that the N1N2-automorphism group is a subgroup of the automorphism group of an AG code. [ABSTRACT FROM AUTHOR]

The groups dVn are an infinite family of groups, first introduced by C. Martínez-Pérez, F. Matucci and B. E. A. Nucinkis, which includes both the Higman-Thompson groups Vn (D 1Vn) and the Brin-Thompson groups nV (D nV2). A description of the groups Aut.Gn;r / (including the groups Gn;1 D Vn) has previously been given by C. Bleak, P. Cameron, Y. Maissel, A. Navas, and F. Olukoya. Their description uses the transducer representations of homeomorphisms of Cantor space introduced in a paper of R. I. Grigorchuk, V.V. Nekrashevich, and V. I. Sushchanskii, together with a theorem of M. Rubin. We generalise the transducers of the latter paper and make use of these transducers to give a description of Aut.dVn/which extends the description of Aut.1Vn/given in the former paper. We make use of this description to show that Out.dV2/Š Out.V2/o Sd, and more generally give a natural embedding of Out.dVn/into Out.Gn;n1 oSd. [ABSTRACT FROM AUTHOR]

In the work by Li (J. Combin. Theory Ser. B , 99 (2009), 447–454.), the author characterized the classification of vertex transitive embeddings of complete graphs, and proposed how to enumerate such maps. In this paper, we study the counting problem of orientable vertex imprimitive complete maps, which is the automorphism group of this map acts imprimitively on its vertex set. Moreover, we obtain the number of non-isomorphic embeddings when the vertex-stabilizer subgroups of the automorphism groups of maps are isomorphic to Z p − 1 with odd prime p. [ABSTRACT FROM AUTHOR]

A family U of non-empty subsets of a set D is called an upfamily if for each set U E U any set F U U belongs to U. The upfamily extension υ(D) of D consists of all upfamilies on D. Any associative binary operation ∗: D×D → D can be extended to an associative binary operation ... In the paper, we show that the upfamily extension (υ(D), -|, -|) of a (strong) doppelsemigroup (D, -|, -|) is a (strong) doppelsemigroup as well and study some properties of this extension. Also we introduce the upfamily functor in the category DSG whose objects are doppelsemigroups and morphisms are doppelsemigroup homomorphisms. We prove that the automorphism group of the upfamily extension of a doppelsemigroup (D, -|, -|) of cardinality |D| ≥ 2 contains a subgroup, isomorphic to C2 × Aut(D, -|, -|). Also we describe the structure of upfamily extensions of all two-element doppelsemigroups and their automorphism groups. [ABSTRACT FROM AUTHOR]

The hull of a linear code plays an important role in determining the complexity of algorithms for checking permutation equivalence of two linear codes and computing the automorphism group of a linear code. Regarding the quantum error correction, linear codes with determined hull are used to construct quantum codes. In this paper, we focus on the hull of Simplex codes and punctured Simplex codes. We firstly study the properties of the matrix produced by the column vectors of a projective space and determine the Euclidean and Hermitian hull of punctured Simplex codes completely. Secondly, we investigate the Euclidean and Hermitian hull of several classes of linear codes from Simplex codes. [ABSTRACT FROM AUTHOR]

In this paper, we give a classification of extremal Type II Z 4 -codes of length 24. This is equivalent to a classification of 4-frames of the Leech lattice up to the action of the automorphism group. [ABSTRACT FROM AUTHOR]

This paper, which is the second of a series of three papers, studies dynamical properties of elements of \operatorname {Out}(F_{\mathtt n}), the outer automorphism group of a nonabelian free group F_{\mathtt n}. We prove that, for every exponentially growing outer automorphism of F_{\mathtt n}, there exists a preferred compact topological space, the space of currents relative to a malnormal subgroup system, on which \phi acts by homeomorphism with a North-South dynamics behavior. [ABSTRACT FROM AUTHOR]

In this paper we settle the question of whether a finite-dimensional vector space V over F p , with p an odd prime, and the family of all the k-sets of elements of V summing up to a given element x, form a 1- (v , k , λ 1) or a 2- (v , k , λ 2) block design, and, in either case, we find a closed form for λ i and characterize the automorphism group. The question is discussed also in the case where the elements of the k-sets are required to be all nonzero, as the two cases happen to be intrinsically inseparable. The "twin case" p = 2 , which has strict connections with coding theory, was completely discussed in a recent paper by G. Falcone and the present author. [ABSTRACT FROM AUTHOR]

Consider the n-dimensional projective space PG (n , q) over a finite field with q elements. A spread in PG (n , q) is a set of lines which partition the point set. A parallelism is a partition of the set of lines by spreads. A deficiency one parallelism is a partial parallelism with one spread less than the parallelism. A transitive deficiency one parallelism corresponds to a parallelism possessing an automorphism group which fixes one spread and is transitive on the remaining spreads. Such parallelisms have been considered in many papers. As a result, an infinite family of transitive deficiency one parallelisms of PG (n , q) has been constructed for odd q, and it has been proved that the deficiency spread of a transitive deficiency one parallelism must be regular, and its automorphism group should contain an elation subgroup of order q 2 . In the present paper we construct parallelisms of PG (3 , 7) invariant under an elation group of order 49 with some additional properties, and thus we succeed to obtain all (46) transitive deficiency one parallelisms of PG (3 , 7) . The three parallelisms from the known infinite family are among them. As a by-product, we also construct a much bigger number (55,022) of parallelisms which have the same spread structure, but are not transitive deficiency one. [ABSTRACT FROM AUTHOR]

In this paper, we study a flag-transitive automorphism group G of symmetric 2- (v , k , λ) designs, where λ > 1 and the order n = k - λ is a prime number. We prove that G is point-primitive, and it is of affine or almost simple type. [ABSTRACT FROM AUTHOR]

A graph is half-arc-transitive if its automorphism group acts transitively on its vertex set, edge set, but not its arc set. In this paper, we study all tetravalent half-arc-transitive graphs of order 12p, where p is a prime. [ABSTRACT FROM AUTHOR]

In this paper, we explore connected Cayley graphs on non-abelian groups of order 5p², where p is a prime greater than 5, and Sylow p-subgroup is cyclic with respect to tetravalent sets that encompass elements with different orders. We prove that these graphs are normal; however, they are not normal edge-transitive, arc-transitive, nor half-transitive. Additionally, we establish that the group is a 5-CI-group. [ABSTRACT FROM AUTHOR]

This paper contributes to the classification of flag-transitive symmetric 2- (v , k , λ) designs with λ prime. We investigate the structure of flag-transitive, point-quasiprimitive automorphism groups (G) of such 2-designs by applying the classification of quasiprimitive permutation groups. It is shown that the automorphism groups (G) have either an abelian socle or a non-abelian simple socle. Moreover, according to the classification of finite simple groups, we demonstrate that point-quasiprimitivity implies point-primitivity of G, except when the socle of G is P S L n (q) . [ABSTRACT FROM AUTHOR]

In this paper, we have found the automorphism group of the Zappa-Szép product of two groups fixing a subgroup. We have computed these automorphisms for a subgroup of order p2 of a group G which is the Zappa-Szép product of two cyclic groups in which one is of order p2 and other is of order m. [ABSTRACT FROM AUTHOR]

This paper is a contribution to the classification of block-transitive 3-designs. Let D = (P , B) be a nontrivial 3- (v , 4 , λ) design and G ≤ A u t (D) acts block-transitively on D with sporadic or alternating socle, then there are exactly 6 incomplete designs as follows: D is isomorphic to a 3- (12 , 4 , λ) design with λ ∈ { 3 , 6 } , and G ≅ M 11 ; D is isomorphic to a 3- (22 , 4 , λ) design with λ ∈ { 3 , 16 } , and S o c (G) = M 22 ; D is isomorphic to a 3-(10, 4, 1) design, and G ≅ M 10 , P G L 2 (9) or P Γ L 2 (9) ; D is isomorphic to a 3-(10, 4, 6) design, and S o c (G) = A 6 . [ABSTRACT FROM AUTHOR]

The Bose-Chaudhuri-Hocquenghem (BCH) codes are a well-studied subclass of cyclic codes that have found numerous applications in error correction and notably in quantum information processing. They are widely used in data storage and communication systems. A subclass of attractive BCH codes is the narrow-sense BCH codes over the Galois field ${\mathrm {GF}}(q)$ with length $q+1$ , which are closely related to the action of the projective general linear group of degree two on the projective line. Despite its interest, not much is known about this class of BCH codes. This paper aims to study some of the codes within this class and specifically narrow-sense antiprimitive BCH codes (these codes are also linear complementary duals (LCD) codes that have interesting practical recent applications in cryptography, among other benefits). We shall use tools and combine arguments from algebraic coding theory, combinatorial designs, and group theory (group actions, representation theory of finite groups, etc.) to investigate narrow-sense antiprimitive BCH Codes and extend results from the recent literature. Notably, the dimension, the minimum distance of some $q$ -ary BCH codes with length $q+1$ , and their duals are determined in this paper. The dual codes of the narrow-sense antiprimitive BCH codes derived in this paper include almost MDS codes. Furthermore, the classification of ${\mathrm {PGL}}(2, p^{m})$ -invariant codes over ${\mathrm {GF}}(p^{h})$ is completed. As an application of this result, the $p$ -ranks of all incidence structures invariant under the projective general linear group ${\mathrm {PGL}}(2, p^{m})$ are determined. Furthermore, infinite families of narrow-sense BCH codes admitting a 3-transitive automorphism group are obtained. Via these BCH codes, a coding-theory approach to constructing the Witt spherical geometry designs is presented. The BCH codes proposed in this paper are good candidates for permutation decoding, as they have a relatively large group of automorphisms. [ABSTRACT FROM AUTHOR]

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This paper presents a criterion for the existence of a birational embedding into a projective plane with non-collinear Galois points for algebraic curves and describes its application via a novel example of a plane curve with non-collinear Galois points. In addition, this paper presents a new characterisation of the Fermat curve in terms of non-collinear Galois points. [ABSTRACT FROM AUTHOR]

Locally repairable codes have been investigated extensively in recent years due to practical applications in distributed storage as well as theoretical interest. However, not much work on asymptotical behavior of locally repairable codes has been done until now. In particular, there is little result on constructive lower bound of asymptotical behavior of locally repairable codes. In this paper, we extend the construction given by Barg et al. via automorphism groups of function field towers. The main advantage of our construction is to allow more flexibility of locality. Furthermore, we show that the Gilbert–Varshamov type bound on locally repairable codes can be improved for all sufficiently large alphabet size $q$. [ABSTRACT FROM AUTHOR]

Let IAn denote the group of IA-automorphisms of a free group of rank n, and let Ibn denote the Torelli subgroup of the mapping class group of an orientable surface of genus n with b boundary components, b D 0; 1. In 1935, Magnus proved that IAn is finitely generated for all n, and in 1983, Johnson proved that Ibn is finitely generated for n ≥ 3. It was recently shown that for each k ∈ N, the k-th terms of the lower central series ykIAn and ykIbn are finitely generated when n ≫-k; however, no information about finite generating sets was known for k > 1. The main goal of this paper is to construct an explicit finite generating set for Y2IAn = [IAn, IAn] and almost explicit finite generating sets for Y2Ibn and the Johnson kernel, which contains Y2Ibn as a finite index subgroup. [ABSTRACT FROM AUTHOR]

A graph is edge-primitive if its automorphism group acts primitively on the edge set of the graph. Edge-primitive graphs form an important subclass of symmetric graphs. In this paper, edge-primitive graphs of order as a product of two distinct primes are completely determined. This depends on non-abelian simple groups with a subgroup of index pq being classified, where p > q are odd primes. [ABSTRACT FROM AUTHOR]

Consider G to be a simple graph with n vertices and m edges, and L (G) to be a Laplacian matrix with Laplacian eigenvalues of μ 1 , μ 2 , ... , μ n = z e r o . Write S k (G) = ∑ i = 1 k μ i as the sum of the k-largest Laplacian eigenvalues of G, where k ∈ { 1 , 2 , ... , n } . The motivation of this study is to solve a conjecture in algebraic graph theory for a special type of graph called a wheel graph. Brouwer's conjecture states that S k (G) ≤ m + k + 1 2 , where k = 1 , 2 , ... , n . This paper proves Brouwer's conjecture for wheel graphs. It also provides an upper bound for the sum of the largest Laplacian eigenvalues for the wheel graph W n + 1 , which provides a better approximation for this upper bound using Brouwer's conjecture and the Grone–Merris–Bai inequality. We study the symmetry of wheel graphs and recall an example of the symmetry group of W n + 1 , n ≥ 3 . We obtain our results using majorization methods and illustrate our findings in tables, diagrams, and curves. [ABSTRACT FROM AUTHOR]

Let G = (V, E) be a connected graph with the vertex-set V and the edgeset E. The subdivision graph S(G) of the graph G is obtained from G by adding a vertex in the middle of every edge of G. In this paper, we investigate some properties of the graphs S (G) and L(S (G)), where L(S (G)) is the line graph of S (G). We will see that S (G) and L(S (G)) inherit some properties of G. For instance, we show that if G Cn, then Aut(G) = Aut(L(S(G))) (as abstract groups), where Cn is the cycle of order n. [ABSTRACT FROM AUTHOR]

In the present paper, we study the outer automorphism groups of the absolute Galois groups of mixed-characteristic local fields from the point of view of anabelian geometry. Let us recall that it is well-known that the natural homomorphism from the automorphism group of a mixed-characteristic local field to the outer automorphism group of the absolute Galois group of the given mixed-characteristic local field is injective. One main result of the present paper is that if the mixed-characteristic local field satisfies certain conditions, then the set of conjugates of the image of this injective homomorphism in the outer automorphism group is infinite, which thus implies that the image of this injective homomorphism is not normal in the outer automorphism group. In particular, one may conclude that it is impossible to establish a functorial group-theoretic reconstruction, from the absolute Galois group, of the "field-theoretic" subgroup, i.e., the image of this injective homomorphism, of the outer automorphism group. [ABSTRACT FROM AUTHOR]

Tang and Ding [IEEE IT 67 (2021) 244-254] studied the class of BCH codes $\mathcal {C}_{(q,q+1,4,1)}$ and their dual codes with $q=2^{m}$ and established that the codewords of the minimum (or the second minimum) weight in these codes support 4-designs or 3-designs. Motivated by this, we further investigate the codewords of the next adjacent weight in such codes and discover more infinite classes of $t$ -designs with $t=3,4$. In particular, we prove that codewords of weight 7 in $\mathcal {C}_{(q,q+1,4,1)}$ support 4-designs for odd $m \geqslant 5$ and they support 3-designs for even $m \geqslant 4$ , which provide infinite classes of simple $t$ -designs with new parameters. Another significant class of $t$ -designs we produce in this paper has complementary designs with parameters 4- $(2^{2s+1}+ 1,5,5)$ ; these designs have the smallest index among all the known simple 4- $(q+1,5,\lambda)$ designs derived from codes for prime powers $q$ ; and they are further proved to be isomorphic to the 4-designs admitting the projective general linear group PGL $(2,2^{2s+1})$ as the automorphism group constructed by Alltop in 1969. [ABSTRACT FROM AUTHOR]

The Euclidean hull of a linear code $C$ is the intersection of $C$ with its Euclidean dual $C^\perp $. The hull with low dimensions gets much interest due to its crucial role in determining the complexity of algorithms for computing the automorphism group of a linear code and for checking permutation equivalence of two linear codes. The Euclidean hull of a linear code has been applied to the so-called entanglement-assisted quantum error-correcting codes (EAQECCs) via classical error-correcting codes. In this paper, we firstly consider linear codes with one-dimensional Euclidean hull from algebraic geometry codes, and then present a general method to construct linear codes with arbitrary dimensional Euclidean hull. Some new EAQECCs are presented. [ABSTRACT FROM AUTHOR]

Many symmetric properties are well-explored in graph theory, especially in graph coloring, such as symmetric graphs defined by the automorphism groups, symmetric drawing of planar graphs, and symmetric functions which are used to count the number of specific colorings of a graph. This paper is devoted to studying the star edge coloring of 1-planar graphs. The star chromatic index χ st ′ (G) of a graph G is defined as the smallest k for which the edges of G can be colored by using k colors so that no two adjacent edges get the same color and no bichromatic paths or cycles of length four are produced. A graph G is called 1-planar if it can be drawn in the plane such that each edge crosses at most one other edge. In this paper, we prove that every 1-planar graph G satisfies χ st ′ (G) ≤ 7.75 Δ + 166 ; and moreover χ st ′ (G) ≤ ⌊ 1.5 Δ ⌋ + 500 if G contains no 4-cycles, and χ st ′ (G) ≤ 2.75 Δ + 116 if G is 3-connected, or optimal, or NIC-planar. [ABSTRACT FROM AUTHOR]

Representations of braid group B n on n ≥ 2 strands by automorphisms of a free group of rank n go back to Artin. In 1991, Kauffman introduced a theory of virtual braids, virtual knots, and links. The virtual braid group V B n on n ≥ 2 strands is an extension of the classical braid group B n by the symmetric group S n . In this paper, we consider flat virtual braid groups F V B n on n ≥ 2 strands and construct a family of representations of F V B n by automorphisms of free groups of rank 2 n . It has been established that these representations do not preserve the forbidden relations between classical and virtual generators. We investigated some algebraic properties of the constructed representations. In particular, we established conditions of faithfulness in case n = 2 and proved that the kernel contains a free group of rank two for n ≥ 3 . [ABSTRACT FROM AUTHOR]

This paper studies the rigidity properties of the abstract commensurator of the outer automorphism group of a universal Coxeter group of rank n, which is the free product Wn of n copies of Z/2Z. We prove that for n ≥ 5 the natural map Out(Wn) → Comm(Out(Wn)) is an isomorphism and that every isomorphism between finite index subgroups of Out(Wn) is given by a conjugation by an element of Out(Wn). [ABSTRACT FROM AUTHOR]

In this article, we introduce a new counting polynomial, namely the orbit polynomial. It is well-known that this polynomial has a unique positive zero δ in the interval [0, 1]. The aim of this paper is to study the specific properties of this polynomial and then determine the location of this root for several classes of complex networks to compare with other graphical measures. Additionally, we compare the unique positive zero measure with several well-known centrality graph measures. [ABSTRACT FROM AUTHOR]

Let S be a finite thick generalized quadrangle, and suppose that G is an automorphism group of S . If G acts primitively on both the points and lines of S , then it is known that G must be almost simple. In this paper, we show that if the socle of G is PSL (2 , q) with q ≥ 4 , then q = 9 and S is the unique generalized quadrangle of order 2. [ABSTRACT FROM AUTHOR]

The notion of hypergraphs, introduced around 1960, is a generalization of that of graphs and one of the initial concerns was to extend some classical results of graph theory. In this paper, we present some connections between hypergraph theory and hypergroup theory. In this regard, we construct two hypergroupoids by defining two new hyperoperations on H, the set of all hypergraphs. We prove that our defined hypergroupoids are commutative hypergroups and we define hyperrings on H by using the two defined hyperoperations. Moreover, we study the fundamental group, complete parts, automorphism group and strongly regular relations of one of our hypergroups. [ABSTRACT FROM AUTHOR]

Locally repairable codes, or locally recoverable codes (LRC for short), are designed for applications in distributed and cloud storage systems. Similar to classical block codes, there is an important bound called the Singleton-type bound for locally repairable codes. In this paper, an optimal locally repairable code refers to a block code achieving this Singleton-type bound. Like classical MDS codes, optimal locally repairable codes carry some very nice combinatorial structures. Since the introduction of the Singleton-type bound for locally repairable codes, people have put tremendous effort into construction of optimal locally repairable codes. There are a few constructions of optimal locally repairable codes in the literature. Most of these constructions are realized via either combinatorial or algebraic structures. In this paper, we apply automorphism group of the rational function field to construct optimal locally repairable codes by considering the group action on projective lines over finite fields. Due to various subgroups of the projective general linear group, we are able to construct optimal locally repairable codes with flexible locality as well as smaller alphabet size comparable to the code length. In particular, we produce new families of $q$ -ary locally repairable codes, including codes of length $q+1$ via cyclic groups. [ABSTRACT FROM AUTHOR]

Let be a compact complex space in Fujiki's class . In this paper, we show that admits a compact Kähler model , that is, there exists a projective bimeromorphic map from a compact Kähler manifold such that the automorphism group lifts holomorphically and uniquely to a subgroup of . As a consequence, we also give a few applications to the Jordan property, the finiteness of torsion groups, and arbitrary large finite abelian subgroups for compact complex spaces in Fujiki's class . [ABSTRACT FROM AUTHOR]

In the present paper, we discuss the injectivity of the natural homomorphism from the automorphism group of a given field to the outer automorphism group of the associated absolute Galois group. We prove that this natural homomorphism is injective in the case where, for instance, the given field may be embedded into the field of fractions of some Noetherian local domain of mixed characteristic. [ABSTRACT FROM AUTHOR]

Let K be the Galois field F q t of order q t , q = p e , p a prime, A = Aut (K) be the automorphism group of K and σ = (σ 0 , ... , σ d - 1) ∈ A d , d ≥ 1 . In this paper the following generalization of the Veronese map is studied: ν d , σ : ⟨ v ⟩ ∈ PG (n - 1 , K) ⟶ ⟨ v σ 0 ⊗ v σ 1 ⊗ ⋯ ⊗ v σ d - 1 ⟩ ∈ PG (n d - 1 , K). Its image will be called the (d , σ) -Veronese variety V d , σ . For d = t , σ a generator of Gal (F q t | F q) and σ = (1 , σ , σ 2 , ... , σ t - 1) , the (t , σ) -Veronese variety V t , σ is the variety studied in [9, 11, 13]. Such a variety is the Grassmann embedding of the Desarguesian spread of PG (n t - 1 , F q) and it has been used to construct codes [3] and (partial) ovoids of quadrics, see [9, 12]. Here, we will show that V d , σ is the Grassmann embedding of a normal rational scroll and any d + 1 points of it are linearly independent. We give a characterization of d + 2 linearly dependent points of V d , σ and for some choices of parameters, V p , σ is the normal rational curve; for p = 2 , it can be the Segre's arc of PG (3 , q t) ; for p = 3 V p , σ can be also a | V p , σ | -track of PG (5 , q t) . Finally, investigate the link between such points sets and a linear code C d , σ that can be associated to the variety, obtaining examples of MDS and almost MDS codes. [ABSTRACT FROM AUTHOR]

We define a four-dimensional Lie algebra g in this paper and then prove that this Lie algebra is solvable but not nilpotent. Due to the fact that g is a Lie algebra, ∀ x , y ∈ g , [ x , y ] = − [ y , x ] , that is, the operation [ , ] has anti symmetry. Symmetry is a very important law, and antisymmetry is also a very important law. We studied the structure of Poisson algebras on g using the matrix method. We studied the necessary and sufficient conditions for the automorphism of this class of Lie algebras, and give the decomposition of its automorphism group by A u t (g) = G 3 G 1 G 2 G 3 G 4 G 7 G 8 G 5 , or A u t (g) = G 3 G 1 G 2 G 3 G 4 G 7 G 8 G 5 G 6 , or A u t (g) = G 3 G 1 G 2 G 3 G 4 G 7 G 8 G 5 G 3 , where G i is a commutative subgroup of A u t (g) . We give some subgroups of g's automorphism group and systematically studied the properties of these subgroups. [ABSTRACT FROM AUTHOR]

A new notion of bent sequence related to Hadamard matrices was introduced recently, motivated by a security application (Solé et al. 2021). In this paper we introduce the analogous notion for complex Hadamard matrices, and we study the self-dual class in length at most 90. We use three competing methods of generation: Brute force, Linear Algebra and Groebner bases. Regular complex Hadamard matrices and Bush-type complex Hadamard matrices provide many examples. We introduce the strong automorphism group of complex Hadamard matrices, which acts on their associated self-dual bent sequences. We give an efficient algorithm to compute that group. We also answer the question which complex Hadamard matrices can be uniquely reconstructed from the off-diagonal elements, define a related concept of mixed-skew Hadamard matrix, and show the existence of mixed-skew Hadamard matrices of small orders. [ABSTRACT FROM AUTHOR]

This paper contributes to the classification of flag-transitive 2-designs. Let D be a 2- (v , k , λ) design with prime square replication number. Suppose that G is a flag-transitive, point-primitive automorphism group of D with socle an alternating group. We prove that D is the unique 2-(10, 6, 5) design with G = A l t (6) or Sym(6). [ABSTRACT FROM AUTHOR]

Regular two-graphs on up to 36 vertices are classified, and recently, the classification of regular two-graphs on 38 and 42 vertices having at least one descendant with a nontrivial automorphism group has been performed. The first unclassified cases are those on 46 and 50 vertices. It is known that there are at least 97 regular two-graphs on 46 vertices leading to 2104 descendants and 54 regular two-graphs on 50 vertices leading to 785 descendants. In this paper, we classified all strongly regular graphs with parameters (45 , 22 , 10 , 11) , (49 , 24 , 11 , 12) , and (50 , 21 , 8 , 9) that have Z 6 as the automorphism group and constructed regular two-graphs from SRGs (45 , 22 , 10 , 11) , SRGs (49 , 24 , 11 , 12) , and SRGs (50 , 21 , 8 , 9) that have automorphisms of order six. In this way, we enumerated all regular two-graphs on up to 50 vertices that have at least one descendant with an automorphism group of order six or at least one strongly regular graph associated with an automorphism group of order six. We found 236 new regular two-graphs on 46 vertices leading to 3172 new SRG (45 , 22 , 10 , 11) and 51 new regular two-graphs on 50 vertices leading to 398 new SRG (49 , 24 , 11 , 12) . [ABSTRACT FROM AUTHOR]

A totally symmetric set is a subset of a group such that every permutation of the subset can be realized by conjugation in the group. The (non-)existence of large totally symmetric sets obstruct homomorphisms, so bounds on the sizes of totally symmetric sets are of particular use. In this paper, we prove that if a group has a totally symmetric set of size fc, it must have order at least (k + 1)!. We also show that with three exceptions, {(1 i) | i = 2...,n} C Sn is the only totally symmetric set making this bound sharp, a fact which gives a new perspective on the automorphism group of Sn. [ABSTRACT FROM AUTHOR]

The hull of a linear code C is the intersection of C with its dual C ⊥ . The hull with low dimensions gets much interest due to its crucial role in determining the complexity of algorithms for computing the automorphism group of a linear code and for checking permutation equivalence of two linear codes. The hull of linear codes has recently found its application to the so-called entanglement-assisted quantum error-correcting codes (EAQECCs). In this paper, we provide a new method to construct linear codes with one-dimensional hull. This construction method improves the code lengths and dimensions of the recent results given by the author. As a consequence, we derive several new classes of optimal linear codes with one-dimensional hull. Some new EAQECCs are presented. [ABSTRACT FROM AUTHOR]

The token graphs of graphs have been studied at least from the 80’s with different names and by different authors. The Johnson graph J(n, k) is isomorphic to the k-token graph of the complete graph Kn. To our knowledge, the unique results about the automorphism groups of token graphs are for the case of the Johnson graphs. In this paper we begin the study of the automorphism groups of token graphs of another graphs. In particular we obtain the automorphism group of the k-token graph of the path graph Pn, for n 6= 2k. Also, we obtain the automorphism group of the 2-token graph of the following graphs: cycle, star, fan and wheel graphs. [ABSTRACT FROM AUTHOR]

The inner automorphism groups of quandles are related to the classification problem of quandles. The inner automorphism group of a quandle is generated by inner automorphisms which are presented by columns in the operation table of the quandle. In this paper, we describe inner automorphisms of an abelian extension of a quandle by expressing columns of the operation table of the extended quandle as columns of the operation table of the original quandle. Such a description will be helpful in studying inner automorphism groups of abelian extensions of quandles. [ABSTRACT FROM AUTHOR]

It was previously shown by Grunewald and Lubotzky that the automorphism group of a free group, Aut(Fn), has a large collection of virtual arithmetic quotients. Analogous results were proved for the mapping class group by Looijenga and by Grunewald, Larsen, Lubotzky, and Malestein. In this paper, we prove analogous results for the automorphism group of a right-angled Artin group for a large collection of defining graphs. As a corollary of our methods we produce new virtual arithmetic quotients of Aut(Fn) for n ≥ 4 where kth powers of all transvections act trivially for some fixed k. Thus, for some values of k, we deduce that the quotient of Aut(Fn) by the subgroup generated by kth powers of transvections contains nonabelian free groups. This expands on results of Malestein and Putman and of Bridson and Vogtmann. [ABSTRACT FROM AUTHOR]

In this paper, we determine all dihedral subgroups in $PGL_{2}(\mathbb {F}_{q})$ , where $q=2^{l}$ and $l$ is a positive integer; and construct binary expurgated and extended Goppa codes with dihedral automorphism groups. Moreover, it is easy to obtain binary quasi-cyclic expurgated or extended Goppa codes. [ABSTRACT FROM AUTHOR]

The Tutte polynomial is an isomorphism invariant of graphs that generalizes the chromatic and the flow polynomials. This two-variable polynomial with integral coefficients is known to carry important information about the properties of the graph. It has been used to prove long-standing conjectures in knot theory. Furthermore, it is related to the Potts and Ising models in statistical physics. The purpose of this paper is to study the interaction between the Tutte polynomial and graph symmetries. More precisely, we prove that if the automorphism group of the graph G contains an element of prime order p, then the coefficients of the Tutte polynomial of G satisfy certain necessary conditions. [ABSTRACT FROM AUTHOR]