Search

A 2- (v , k , λ) design is additive (or strongly additive) if it is possible to embed it in a suitable abelian group G in such a way that its block set is contained in (or coincides with) the set of all zero-sum k-subsets of its point set. Explicit results on the additivity or strong additivity of symmetric designs and subspace 2-designs are presented. In particular, the strong additivity of PG d (n , q) , which was known to be additive only for q = 2 or d = n - 1 , is always established. [ABSTRACT FROM AUTHOR]

The symmetric 2- (v , k , λ) designs with k > λ (λ − 3) / 2 admitting a flag-transitive point-imprimitive automorphism group are completely classified: they are the known 2-designs with parameters (16 , 6 , 2) , (45 , 12 , 3) , (15 , 8 , 4) or (96 , 20 , 4). [ABSTRACT FROM AUTHOR]

Let a = (a i) i ≥ 1 be a sequence in a field F , and f : F × F → F be a function such that f (a i , a i) ≠ 0 for all i ≥ 1. For any tournament T over [ n ] , consider the n × n symmetric matrix M T with zero diagonal whose (i , j) th entry (for i < j) is f (a i , a j) if i → j in T , and f (a j , a i) if j → i in T. It is known [3] that if T is a uniformly random tournament over [ n ] , then rank (M T) ≥ (1 2 − o (1)) n with high probability when char (F) ≠ 2 and f is a linear function. In this paper, we investigate the other extremal question: how low can the ranks of such matrices be? We work with sequences a that take only two distinct values, so the rank of any such n × n matrix is at least n / 2. First, we show that the rank of any such matrix depends on whether an associated bipartite graph has certain eigenvalues of high multiplicity. Using this, we show that if f is linear, then there are real matrices M T (f ; a) of rank at most n 2 + O (1). For rational matrices, we show that for each ε > 0 we can find a sequence a (ε) for which there are matrices M T (f ; a) of rank at most (1 2 + ε) n + O (1). These matrices are constructed from symmetric designs, and we also use them to produce bisection-closed families of size greater than ⌊ 3 n / 2 ⌋ − 2 for n ≤ 15 , which improves the previously best known bound given in [5]. [ABSTRACT FROM AUTHOR]

Complementary pairs of symmetric 2-designs are equivalent to coherent configurations of type (2, 2; 2). D. G. Higman studied these coherent configurations and adjacency algebras of coherent configurations over a field of characteristic zero. These are always semisimple. We investigate these algebras over fields of any characteristic prime and the structures. [ABSTRACT FROM AUTHOR]

In 2022, the second author found a prolific construction of strongly regular graphs, which is based on joining a coclique and a divisible design graph with certain parameters. The construction produces strongly regular graphs with the same parameters as the complement of the symplectic graph Sp (2 d , q) . In this paper, we determine the parameters of strongly regular graphs which admit a decomposition into a divisible design graph and a coclique attaining the Hoffman bound. In particular, it is shown that when the least eigenvalue of such a strongly regular graph is a prime power, its parameters coincide with those of the complement of Sp (2 d , q) . Furthermore, a generalization of the construction is discussed. [ABSTRACT FROM AUTHOR]

In this article, we study symmetric designs admitting flag‐transitive, point‐imprimitive almost simple automorphism groups with socle sporadic simple groups. As a corollary, we present a classification of symmetric designs admitting flag‐transitive automorphism group whose socle is a sporadic simple group, and in conclusion, there are exactly seven such designs, one of which admits a point‐imprimitive automorphism group and the remaining are point‐primitive. [ABSTRACT FROM AUTHOR]

The study of regular incidence structures, such as projective planes and symmetric block designs, is a well established topic in discrete mathematics. The work of Bruck, Ryser and Chowla in the mid-twentieth century applied the Hasse-Minkowski local-global theory for quadratic forms to derive non-existence results for certain design parameters. Several combinatorialists have provided alternative proofs of this result, replacing conceptual arguments with algorithmic ones. In this paper, we show that the methods required are purely linear-algebraic and are no more difficult conceptually than the theory of the Jordan Canonical Form. Computationally, they are rather easier. We conclude with some classical and recent applications to design theory, including a novel application to the decomposition of incidence matrices of symmetric designs. [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]

This paper studies symmetric (v , k , λ) designs D with v odd and admitting a block-transitive automorphism group G whose socle is an alternating group A n . We indeed show that for given λ , only finitely many such designs exist and prove that if 2 ≤ λ ≤ 5 , then D = PG 2 (3 , 2) or its complement and G = S 5 , S 6 , A n (n = 5 , 6 , 7 , 8) . [ABSTRACT FROM AUTHOR]

Let r1 ${r}_{1}$ and r2,(r1>r2) ${r}_{2},({r}_{1}\gt {r}_{2})$ be the two replication numbers of a λ $\lambda $‐design D $D$. We denote the block size ∣Bj∣ $| {B}_{j}| $ by kj ${k}_{j}$ and by kj′ ${k}_{j}^{^{\prime} }$ (respectively, kj* ${k}_{j}^{* }$) the number of points with replication number r1 ${r}_{1}$ (respectively, r2 ${r}_{2}$) in block Bj ${B}_{j}$ of D $D$. Take g=gcdr1−r2gcd(r1−1,r2−1),λ,λ=λ1g,r1−r2gcd(r1−1,r2−1)=gm $g=\text{gcd}\left(\frac{{r}_{1}-{r}_{2}}{\text{gcd}({r}_{1}-1,{r}_{2}-1)},\lambda \right),\lambda ={\lambda }_{1}g,\frac{{r}_{1}-{r}_{2}}{\text{gcd}({r}_{1}-1,{r}_{2}-1)}=gm$, for positive integers λ1,m ${\lambda }_{1},m$ and let g1 ${g}_{1}$ be the largest divisor of λ1 ${\lambda }_{1}$ such that if p $p$ is a prime dividing g1 ${g}_{1}$, then p $p$ divides g $g$. We obtain the following results: 1.If D $D$ is a type‐1 λ $\lambda $‐design then kj′ ${k}_{j}^{^{\prime} }$ and kj* ${k}_{j}^{* }$ are divisible by gg1 $g{g}_{1}$.2.If kj′−kj* ${k}_{j}^{^{\prime} }-{k}_{j}^{* }$ is divisible by gg1 $g{g}_{1}$ for all blocks Bj ${B}_{j}$ of a design D $D$, then D $D$ is a type‐1 λ $\lambda $‐design. [ABSTRACT FROM AUTHOR]

Let D = (P , B) be a symmetric 2- (v , k , λ) design admitting a flag-transitive, point-imprimitive automorphism group G that leaves invariant a non-trivial partition Σ of P. Praeger and Zhou [42] have shown that, there is a constant k 0 such that, for each B ∈ B and Δ ∈ Σ , the size of | B ∩ Δ | is either 0 or k 0. In the present paper we show that, if k > λ (λ − 3) / 2 and k 0 ⩾ 3 , D is isomorphic to one of the known flag-transitive, point-imprimitive symmetric 2-designs with parameters (45 , 12 , 3) or (96 , 20 , 4). [ABSTRACT FROM AUTHOR]

In the classical probability model, let f (n) be the maximum number of pairwise independent events for the sample space with n sample points. The determination of f (n) is equivalent to the problem of determining the maximum cardinality of specific intersecting families on the set { 1 , 2 , ... , n }. We show that f (n) ≤ n + 1 , and f (n) = n + 1 if there exists a Hadamard matrix of order n. [ABSTRACT FROM AUTHOR]

Coded distributed computing (CDC) is an efficient method to reduce the communication load in general distributed computing frameworks such as MapReduce. In these systems, one usually needs to split the data set into disjoint files and design several output functions to complete a computational task. Li et al. provided some CDC schemes achieving optimal communication load. However, as the number of computing nodes increases, the numbers of input files and output functions of such schemes grow too fast to be applied in practice. In this paper, several infinite families of cascaded CDC schemes, where each output function is computed multiple times, are constructed by taking advantage of symmetric designs. Most importantly, the numbers of input files and output functions of each new scheme are linear with the number of computing nodes and the communication load of each new scheme approximates to that of the CDC scheme derived by Li et al. when the number of nodes becomes large. [ABSTRACT FROM AUTHOR]

In this article, we prove that if a nontrivial symmetric (v, k, λ) design admit a flag-transitive and point-primitive automorphism group G, then the socle X of G cannot be a projective special unitary group of dimension five. As a corollary, we list all exist nineteen non-isomorphism such designs in which λ ∈ {1, 2, 3, 4, 6, 12, 16, 18} and X = PSUn(q) with (n, q) ∈ {(2, 7),(2, 9),(2, 11),(3, 3),(4, 2)}. [ABSTRACT FROM AUTHOR]

In this article, we investigate 2- (v , k , λ) designs with gcd (r , λ) = 1 admitting flag-transitive automorphism groups G. We prove that if G is an almost simple group, then such a design belongs to one of the eight infinite families of 2-designs or it is one of the eleven well-known examples. We describe all these examples of designs. We, in particular, prove that if D is a symmetric (v , k , λ) design with gcd (k , λ) = 1 admitting a flag-transitive automorphism group G, then either G ⩽ A Γ L 1 (q) for some odd prime power q, or D is the point-hyperplane design or the unique Hadamard design with parameters (11, 5, 2). [ABSTRACT FROM AUTHOR]

Symetric designs are interesting objects of combinatorics, and have some relations with coding theory, difference sets, geometry and finite group theory. They have applications on statistics and design experiments. In the present paper we study an existing problem for symmetric design due to Bruck, Ryser and Chowla and write an algorithm by using their theorem called BRC Theorem. [ABSTRACT FROM AUTHOR]

Let X = (V , E) be a finite regular graph and H t (u , v) , u , v ∈ V , the heat kernel on X. We prove that, if the graph X is bipartite and has four distinct Laplacian eigenvalues, the ratio H t (u , v) / H t (u , u) , u , v ∈ V , is monotonically non-decreasing as a function of t. The key to the proof is the fact that such a graph is an incidence graph of a symmetric 2-design. [ABSTRACT FROM AUTHOR]

A Ryser design D on v points is a collection of v proper subsets (called blocks) of a point-set with v points such that every two blocks intersect each other in λ points (and λ < v is a fixed number) and there are at least two block sizes. A design D is called a symmetric design, if every point of D has the same replication number (or equivalently, all the blocks have the same size) and every two blocks intersect each other in λ points. The only known construction of a Ryser design is via block complementation of a symmetric design. Such a Ryser design is called a Ryser design of Type-1. This is the ground for the Ryser–Woodall conjecture: "every Ryser design is of Type-1". This long standing conjecture has been shown to be valid in many situations. Let D denote a Ryser design of order v, index λ and replication numbers r 1 , r 2 . Let e i denote the number of points of D with replication number r i (with i = 1 , 2 ). Call a block A of D small (respectively large) if | A | < 2 λ (respectively | A | > 2 λ ) and average if | A | = 2 λ . Let D denote the integer e 1 - r 2 and let ρ > 1 denote the ratio of r 1 - 1 and r 2 - 1 . Main results of the present article are the following: An equivalence relation on the set of Ryser designs is established. Some observations on the block complementation procedure of Ryser–Woodall are made. It is shown that a Ryser design with two block sizes one of which is an average block size is of Type-1. It is also shown that, under the assumption that large and small blocks do not coexist in any Ryser design equivalent to a given Ryser design, the given Ryser design must be of Type-1. [ABSTRACT FROM AUTHOR]

In this paper we propose unreduced balanced incomplete block designs which are multiple balanced incomplete block designs of resolvable type designs with no repeated blocks and also have quasi - symmetric structure with k intersection numbers between the blocks with illustrations. [ABSTRACT FROM AUTHOR]

Let D be a 2- (v , k , λ) symmetric design with k and λ prime powers. If D admits a flag-transitive, point-imprimitive automorphism group G, we show that both k and λ must be powers of 2. Moreover, there exists an integer m such that either (a) D has parameters (v , k , λ) = (2 2 m + 2 - 1 , 2 2 m + 1 , 2 2 m) , and G preserves a partition of the points into 2 m + 1 + 1 classes of size 2 m + 1 - 1 , or (b) D has parameters (v , k , λ) = ((2 2 m - 1 - 2 m + 1) (2 m - 1 + 1) , 2 2 m - 1 , 2 m) , and G preserves a partition of the points into 2 2 m - 1 - 2 m + 1 classes of size 2 m - 1 + 1 . [ABSTRACT FROM AUTHOR]

A (v , k , λ) symmetric design is said to have the symmetric difference property (SDP) if the symmetric difference of any three blocks is either a block or the complement of a block. The designs associated to the symplectic difference sets introduced by Kantor (J Algebra 33:43–58, 1975) have the SDP. Parker (J Comb Theory Ser A 67:23–43, 1994) claimed that the symplectic design on 64 points is the only SDP design on 64 points admitting an abelian regular automorphism group (an abelian difference set). We show in this paper that there is an SDP design on 64 points that is not isomorphic to the symplectic design and yet admits the group C 8 × C 4 × C 2 as a regular automorphism group. This abelian difference set is the first in an infinite family of abelian difference sets whose designs have the SDP and yet are not isomorphic to the symplectic designs of the same order. We define a new method for establishing the non-isomorphism of the two families. [ABSTRACT FROM AUTHOR]

Let r and r′, (r > r′) be the two replication numbers of a λ‐design D. We show that if r−r′ is a prime power or 33, then D is a design of type‐1. We derive two inequalities, where equality holds if and only if D is a design of type‐1. Let D be a λ‐design with two block sizes with v=np+1 points, where p is prime and 2⩽n⩽22. We develop a procedure to obtain exceptions for such type‐1 λ‐designs, refining earlier results. [ABSTRACT FROM AUTHOR]

This study presents a wideband crossover developed using substrate integrated coaxial line (SICL) technology. The proposed SICL‐based crossover isolates the two physically overlapping channels by routing the signal in a middle conductor of SICL through the transmission line created in the top and bottom ground plane using a metalised blind via. An equivalent model is proposed to study the role of substrate height in the proposed crossover. The design methodology is affirmed by fabricating and testing the experimental prototype. The measured results show isolation between channels are more than 19 dB with better than 18 dB return loss and less than 1.2 dB insertion loss from 0.2 to 20 GHz. Owing to the non‐dispersive nature of SICL line and simplistic design technique with minimal discontinuities the designed crossover demonstrates low peak‐to‐peak group delay variation, less than 0.04 ns over the entire bandwidth. The proposed crossover due to its symmetric design provides low path difference of less than ±2.45° up to 20 GHz (≤±1° up to 10 GHz) without any additional phase compensation technique. Utilization of electromagnetically robust self‐packaged components designed in SICL technology provides superior front‐end communication system within a small footprint. [ABSTRACT FROM AUTHOR]

The dual codes of the ternary linear codes of the residual designs of biplanes on 56 points are used to prove the nonexistence of quasisymmetric 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. [ABSTRACT FROM AUTHOR]

In this article, we study symmetric (v,k,λ) designs admitting a flag‐transitive and point‐primitive automorphism group G whose socle is isomorphic to a projective special linear group of dimension at most four. We, in particular, determine all such possible parameters (v,k,λ) and show that such a design belongs to one of two infinite families of point‐hyperplane designs or it is isomorphic to a design with parameters (7, 3, 1), (7, 4, 2), (11, 5, 2), (11, 6, 3), (15, 8, 4), (35, 18, 9), or (56, 45, 18). [ABSTRACT FROM AUTHOR]

Abstract In this article, we study symmetric (v , k , λ) designs admitting a flag-transitive and point-primitive automorphism group G whose socle is PSU 4 (q). We prove that there exist eight non-isomorphic such designs for which λ ∈ { 3 , 6 , 18 } and G is either PSU 4 (2) , or PSU 4 (2) : 2. [ABSTRACT FROM AUTHOR]

A linking system of difference sets is a collection of mutually related group difference sets, whose advantageous properties have been used to extend classical constructions of systems of linked symmetric designs. The central problems are to determine which groups contain a linking system of difference sets, and how large such a system can be. All previous constructive results for linking systems of difference sets are restricted to 2‐groups. We use an elementary projection argument to show that neither the McFarland/Dillon nor the Spence construction of difference sets can give rise to a linking system of difference sets in non‐2‐groups. We make a connection to Kerdock and bent sets, which provides large linking systems of difference sets in elementary abelian 2‐groups. We give a new construction for linking systems of difference sets in 2‐groups, taking advantage of a previously unrecognized connection with group difference matrices. This construction simplifies and extends prior results, producing larger linking systems than before in certain 2‐groups, new linking systems in other 2‐groups for which no system was previously known, and the first known examples in nonabelian groups. [ABSTRACT FROM AUTHOR]

Let G be a flag-transitive automorphism group of a symmetric (v , k , λ) design D with k > λ (λ − 2). O'Reilly Regueiro proved that if G is point-imprimitive, then D has parameters (v , k , λ) = (λ 2 (λ + 2) , λ (λ + 1) , λ). In the present paper, we consider the case that G is point-primitive. By applying the O'Nan-Scott Theorem, we prove that G must be of affine type or almost simple type. [ABSTRACT FROM AUTHOR]

Wireless sensor networks as the key infrastructure of the new networking paradigm are vulnerable against different kinds of attacks. Therefore, ensuring a secure communication between the sensor nodes is important. One of the most critical issues in this regard is the key distribution mechanism. Due to the random deployment of the sensors in the target area, key pre-distribution is a promising approach, in which a list of keys, so-called key-ring, is pre-distributed to each sensor node before deployment. To establish a secure communication, two nodes must share a common key from their key-rings. In this paper, we consider a hybrid key pre-distribution approach based on the symmetric design. We propose a new scheme, which is a modification of the hybrid symmetric design in order to improve the connectivity and resilience. Considering the trade-off between resilience and connectivity, we introduce a new parameter based on the application requirement. The experimental results and analytical analysis approve the efficiency of our proposed approach and introduced parameter. [ABSTRACT FROM AUTHOR]

A resolving set for a graph Γ is a collection of vertices S , chosen so that for each vertex v , the list of distances from v to the members of S uniquely specifies v . The metric dimension μ ( Γ ) is the smallest size of a resolving set for Γ . We consider the metric dimension of two families of incidence graphs: incidence graphs of symmetric designs, and incidence graphs of symmetric transversal designs (i.e. symmetric nets). These graphs are the bipartite distance-regular graphs of diameter 3, and the bipartite, antipodal distance-regular graphs of diameter 4, respectively. In each case, we use the probabilistic method in the manner used by Babai to obtain bounds on the metric dimension of strongly regular graphs, and are able to show that μ ( Γ ) = O ( n log n ) (where n is the number of vertices). [ABSTRACT FROM AUTHOR]

A binary [98280,24,47104]2 projective two-weight code related to the sporadic simple group Co1 of Conway is constructed as a faithful and absolutely irreducible submodule of the permutation module induced by the primitive action of Co1 on the cosets of Co2. The dual code of this code is a uniformly packed [98280,98256,3]2 code. The geometric significance of the codewords of the code can be traced to the vectors in the Leech lattice, thus revealing that the stabilizer of any non-zero weight codeword in the code is a maximal subgroup of Co1. Similarly, the stabilizer of the codewords of minimum weight in the dual code is a maximal subgroup of Co1. As by-product, a new strongly regular graph on 16777216 vertices and valency 98280 is constructed using the codewords of the code. [ABSTRACT FROM AUTHOR]

For any positive integer m , the complete graph on 2 2 m ( 2 m + 2 ) vertices is decomposed into 2 m + 1 commuting strongly regular graphs, which give rise to a symmetric association scheme of class 2 m + 2 − 2 . Furthermore, the eigenmatrices of the symmetric association schemes are determined explicitly. As an application, the eigenmatrix of the commutative strongly regular decomposition obtained from the strongly regular graphs is derived. [ABSTRACT FROM AUTHOR]

Let be a subgroup of the full automorphism group of a - symmetric design . If is flag-transitive and point-primitive, then Soc cannot be , , and . [ABSTRACT FROM AUTHOR]

We show that there are exactly 4285 symmetric (45,12,3) designs that admit nontrivial automorphisms. Among them there are 1161 self-dual designs and 1562 pairs of mutually dual designs. We describe the full automorphism groups of these designs and analyze their ternary codes. R. Mathon and E. Spence have constructed 1136 symmetric (45,12,3) designs with trivial automorphism group, which means that there are at least 5421 symmetric (45,12,3) designs. Further, we discuss trigeodetic graphs obtained from the symmetric (45; 12; 3) designs. We prove that k-geodetic graphs constructed from mutually non-isomorphic designs are mutually non-isomorphic, hence there are at least 5421 mutually non-isomorphic trigeodetic graphs obtained from symmetric (45; 12; 3) designs. [ABSTRACT FROM AUTHOR]

The optimal designs for Fourier regression models under the D-optimality criterion are discussed in this article. First, we investigate the D-optimal designs for estimating two coefficients corresponding to either sine or cosine terms in a full Fourier regression model. In many biological applications, estimating such specific pairs of coefficients is of interest. As a result of this article, the D-optimal designs for estimating these 'coefficient pairs' can be constructed either explicitly or numerically for Fourier regression models with any order. Our resulting designs are provided for Fourier regression models with order less than 6. Secondly, we discuss the sensitivity of our resulting optimal designs for a full Fourier regression model when the true model is actually a reduced version of the assumed one. Lastly, we provide the algorithm for obtaining the D-optimal designs for a reduced Fourier regression model and the D-optimal designs for a useful reduced Fourier model are constructed. The comparison study shows that the constructed designs incorporating the reduced model are efficient. [ABSTRACT FROM AUTHOR]

Let G be an automorphism group of a 2-(v, k, 4) symmetric design Ɗ. In this paper, we prove that if G is flag-transitive point-primitive, then the socle of G cannot be an exceptional group of Lie type. [ABSTRACT FROM AUTHOR]

Our aim is to study the modular representation theory of coherent configurations. Let p be a prime. We consider structures of modular adjacency algebras of coherent configurations obtained from combinatorial designs. The structures of standard modules of modular adjacency algebras provide more information than p -ranks of incidence matrices of combinatorial designs. [ABSTRACT FROM AUTHOR]

It is shown that, if [ABSTRACT FROM AUTHOR]

In this article, we study symmetric (v , k , λ) designs D with λ ⩽ 10 admitting flag-transitive and point-primitive almost simple automorphism group G with socle X. We, in particular, determine all such possible designs and show that such a design is isomorphic to a Desarguesian plane with parameter set (q 2 + q + 1 , q + 1 , 1) and X = PSL 3 (q) , or D is isomorphic to one of the seventeen well-known designs. [ABSTRACT FROM AUTHOR]