In this paper we investigate combinatorial constructions for w -cyclic holey group divisible packings with block size three (3-HGDPs). For any positive integers u , v , w with u ≡ 0 , 1   ( mod 3 ) , the exact number of base blocks of a maximum w -cyclic 3-HGDP of type ( u , w v ) is determined. This result is used to determine the exact number of codewords in a maximum three-dimensional ( u × v ×

A k -extended q -near Skolem sequence of order n , denoted by N n q ( k ) , is a sequence s 1 , s 2 , … , s 2 n − 1 , where s k = 0 and for each integer ℓ ∈ [ 1 , n ] \ { q } there are two indices i , j such that s i = s j = ℓ and ∣ i − j ∣ = ℓ . For a N n q ( k ) to exist it is necessary that q ≡ k ( mod 2 ) when n ≡ 0 , 1 ( mod 4 ) and q ≢ k ( mod 2 ) when n ≡ 2 , 3 ( mod 4 ) , and it is also necessary

An m × n row-column factorial design is an arrangement of the elements of a factorial design into a rectangular array. Such an array is used in experimental design, where the rows and columns can act as blocking factors. Formally, for any integer q , let [ q ] = { 0 , 1 , … , q − 1 } . The q k (full) factorial design with replication α is the multiset consisting of α occurrences of each element of

A strongly regular decomposition of a strongly regular graph is a partition of the vertex set into two parts on which the induced subgraphs are strongly regular, or cliques or cocliques. Strongly regular designs (srd's) as defined by D. G. Higman are coherent configurations of rank 10 with two fibers in which the homogeneous configuration on each fiber is a strongly regular graph. Haemers and Higman

An ( n , k ) -Sperner partition system is a set of partitions of some n -set such that each partition has k nonempty parts and no part in any partition is a subset of a part in a different partition. The maximum number of partitions in an ( n , k ) -Sperner partition system is denoted SP ( n , k ) . In this paper we introduce a new construction for Sperner partition systems based on a division of the

We investigate p -ary t -designs which are simultaneously designs for all t , which we call universal p -ary designs. Null universal designs are well understood due to Gordon James via the representation theory of the symmetric group. We study non-null designs and determine necessary and sufficient conditions on the coefficients for such a design to exist. This allows us to classify all universal designs

The subject of this paper is the study of small complete arcs in PG ( 2 , q ) , for q odd, with at least ( q + 1 ) ∕ 2 points on a conic. We give a short comprehensive proof of the completeness problem left open by Segre in his seminal work. This gives an alternative to Pellegrino's long proof which was obtained in a series of papers in the 1980s. As a corollary of our analysis, we obtain a counterexample

This paper studies flag-transitive 2- ( v , k , λ ) designs with ( r , λ ) = 2 . We prove that if D is a flag-transitive nontrivial 2- ( v , k , λ ) design with ( r , λ ) = 2 , then G is of affine or almost simple type. Furthermore, we analyze the flag-transitive automorphism groups of almost simple type with S o c ( G ) = A n for n ≥ 5 , and obtain that, up to isomorphism, there are 11 designs.

We construct new resolvable decompositions of complete multigraphs and complete equipartite multigraphs into cycles of variable lengths (and a perfect matching if the vertex degrees are odd). We develop two techniques: layering, which allows us to obtain 2-factorizations of complete multigraphs from existing 2-factorizations of complete graphs, and detachment, which allows us to construct resolvable

In this paper, we first study a general type of affine-invariant codes and their support 2-designs. Then we derive infinite families of 2-designs from a special class of affine-invariant codes and obtain their parameters by determining the weight distribution of the linear codes.

A Latin array is a matrix of symbols in which no symbol occurs more than once within a row or within a column. A diagonal of an n × n array is a selection of n cells taken from different rows and columns of the array. The weight of a diagonal is the number of different symbols on it. We show via computation that every Latin array of order n ⩽ 11 has a diagonal of weight at least n − 1 . A corollary

The codewords of weight 10 of the [42, 21, 10] extended binary quadratic residue code are shown to hold a design of parameters 3 − ( 42 , 10 , 18 ) . Its automorphism group is isomorphic to P S L ( 2 , 41 ) . Its existence can be explained neither by a transitivity argument, nor by the Assmus–Mattson theorem.

Let V denote an r -dimensional vector space over F q n , the finite field of q n elements. Then V is also an r n -dimension vector space over F q . An F q -subspace U of V is ( h , k ) q -evasive if it meets the h -dimensional F q n -subspaces of V in F q -subspaces of dimension at most k . The ( 1 , 1 ) q -evasive subspaces are known as scattered and they have been intensively studied in finite geometry

In 1987, Huw Davies proved that, for a flag-transitive point-imprimitive 2- ( v , k , λ ) design, both the block-size k and the number v of points are bounded by functions of λ , but he did not make these bounds explicit. In this paper we derive explicit polynomial functions of λ bounding k and v . For λ ⩽ 4 we obtain a list of “numerically feasible” parameter sets v , k , λ together with the number

In this article, we continue to study the existence of large sets of partitioned incomplete Latin squares (LSPILS). We complete the determination of the spectrum of an LSPILS ( g n ) and prove that there exists an LSPILS ( g n ) if and only if g ≥ 1 , n ≥ 3 , and ( g , n ) ≠ ( 1 , 6 ) . We also start the investigation of LSPILS with two group sizes and prove that there exists an LSPILS + ( g n ( 2

It is well known that, whenever k divides n , the complete k ‐uniform hypergraph on n vertices can be partitioned into disjoint perfect matchings. Equivalently, the set of k ‐subsets of an n‐set can be partitioned into parallel classes so that each parallel class is a partition of the n ‐set. This result is known as Baranyai's theorem, which guarantees the existence of Baranyai partitions. Unfortunately

A partial Room square is maximal if no further pair of elements can be placed into any unoccupied cell without violating the conditions that define a partial Room square. This article is concerned with determining the spectrum of volumes of maximal partial Room squares of order n where the volume is the number of occupied cells and the order n is even.

A Γ ‐magic rectangle set M R S Γ ( a , b ; c ) of order a b c is a set of c arrays of size ( a × b ) whose entries are elements of a finite Abelian group Γ of order a b c , each appearing once, with all row sums in each rectangle equal to a constant ω ∈ Γ and all column sums in each rectangle equal to a constant δ ∈ Γ . There is known a complete characteristic of a MRS Γ ( a , b ; c ) for { a , b }

We consider the problem of existence of perfect 2‐colorings (equitable 2‐partitions) of Hamming graphs with given parameters. We start with conditions on parameters of graphs and colorings that are necessary for their existence. Next we observe known constructions of perfect colorings and propose some new ones giving new parameters. At last, we deduce which parameters of colorings are covered by these

The study of symmetric configurations v 3 with block size 3 has a long and rich history. In this paper we consider two colouring problems which arise naturally in the study of these structures. The first of these is weak colouring, in which no block is monochromatic; the second is strong colouring, in which every block is multichromatic. The former has been studied before in relation to blocking sets

In this note, we present a corrected proof of Lemma 5.8 which was wrong due to the incorrectness of Construction 2.3, from our paper.

In this paper, we focus on partial geometric designs with block sizes three and four. First, we show that a partial geometric design with block size three is a 2‐design, or a transversal design, or each line of a generalized quadrangle that is repeated λ times. Second, we introduce the concept of type of partial geometric designs with block size four and characterize partial geometric designs of type

A pentagonal geometry PENT ( k , r ) is a partial linear space, where every line, or block, is incident with k points, every point is incident with r lines, and for each point x , there is a line incident with precisely those points that are not collinear with x . An opposite line pair in a pentagonal geometry consists of two parallel lines such that each point on one of the lines is not collinear

Consider the usual multiplication table n × n in which all entries strictly greater than n were deleted. Can we fill in the empty places so that the resulting table will become a multiplication table of a group? This intriguing question arose in connection to the famous Graham's Greatest Common Divisor Problem. We prove here a weaker statement, namely, that one may always complete such a partial multiplication

We construct some large families of permutations of Z n such that the sum of any two permutations from a family is again a permutation of Z n (not necessarily in the same family). Our families are significantly larger than the largest such families known so far in the literature (depending on the value of n , the improvement can be as large as exponential). We also show that our families are maximal

Using group rings and characters as in the theory of abelian difference sets, some nonexistence results for strong difference families are provided. Existences of strong difference families with base block size 3 ≤ k ≤ 9 are discussed. Via strong difference families, eight 2‐ ( v , k , λ ) designs whose existences were unknown are constructed for ( v , k , λ ) ∈ { ( 3417 , 8 , 1 ) , ( 3753 , 8 , 1

An r‐golf design of order v , briefly by r‐G(v), is a large set of idempotent Latin squares of order v (ILS ( v ) s) which contains r symmetric ILS ( v ) s and v − r − 2 2 transposed pairs of ILS ( v ) s. In this paper, we mainly consider the existence problem of r‐G(v)s. We present several recursive constructions and also display some direct constructions. As an application, several infinite classes

Let n and ℓ be positive integers. Recent papers by Kreher, Stinson, and Veitch have explored variants of the problem of ordering the points in a triple system (such as a Steiner triple system [STS], directed triple system, or Mendelsohn triple system) on n points so that no block occurs in a segment of ℓ consecutive entries (thus the ordering is locally block‐avoiding). We describe a greedy algorithm

A technique for classifying Butson‐type Hadamard matrices over the complex 4th roots of unity is developed. The technique resembles the approach used by Kharaghani and Tayfeh‐Rezaie in their classification of the (real) 32 × 32 Hadamard matrices, where the search is split into parts according to matrix types. A classification of the 18 × 18 Butson‐type Hadamard matrices over the complex 4th roots of

A ( k , l ) ‐configuration is a set of l blocks on k points. For Steiner triple systems, ( l + 2 , l ) ‐configurations are of particular interest. The smallest nontrivial such configuration is the Pasch configuration, which is a ( 6 , 4 ) ‐configuration. A Steiner triple system of order v , an STS ( v ) , is r ‐sparse if it does not contain any ( l + 2 , l ) ‐configuration for 4 ≤ l ≤ r . The existence

Let K n + I denote the complete graph of even order with a 1‐factor duplicated. The spouse‐loving variant of the Oberwolfach Problem, denoted O P + ( m 1 , m 2 , … , m t ) , asks for the existence of a 2‐factorization of K n + I in which each 2‐factor consists of cycles of length m i , for all i , 1 ≤ i ≤ t , such that n = m 1 + m 2 + ⋯ + m t . If m 1 = m 2 = ⋯ = m t = m , then the problem is denoted

Erdős and Lovasz noticed that an $r$-uniform intersecting hypergraph $H$ with maximal covering number, that is $\tau(H)=r$, must have at least $\frac{8}{3}r-3$ edges. There has been no improvement on this lower bound for 45 years. We try to understand the reason by studying some small cases to see whether the truth lies very close to this simple bound. Let $q(r)$ denote the minimum number of edges

We study the localization number of incidence graphs of designs. In the localization game played on a graph, the cops attempt to determine the location of an invisible robber via distance probes. The localization number of a graph $G$, written $\zeta(G)$, is the minimum number of cops needed to ensure the robber's capture. We present bounds on the localization number of incidence graphs of balanced

Let PG$(\mathbb{F}_q^v)$ be the $(v-1)$-dimensional projective space over $\mathbb{F}_q$ and let $\Gamma$ be a simple graph of order ${q^k-1\over q-1}$ for some $k$. A 2$-(v,\Gamma,\lambda)$ design over $\mathbb{F}_q$ is a collection $\cal B$ of graphs (blocks) isomorphic to $\Gamma$ with the following properties: the vertex-set of every block is a subspace of PG$(\mathbb{F}_q^v)$; every two distinct

It is shown that for $v\ne 7,8,11,12$ or $13$, there exists an optimal covering with triples on $v$ points that contains no Pasch configurations.

An oriented tetrahedron defined on four vertices is a set of four cyclic triples with the property that any ordered pair of vertices is contained in exactly one of the cyclic triples. A tetrahedral quadruple system of order n with index λ , denoted by TQS λ ( n ) , is a pair ( X , ℬ ) , where X is an n ‐set and ℬ is a set of oriented tetrahedra (blocks) such that every cyclic triple on X is contained

A $k$-cycle with a pendant edge attached to each vertex is called a $k$-sun. The existence problem for $k$-sun decompositions of $K_v$, with $k$ odd, has been solved only when $k=3$ or $5$. By adapting a method used by Hoffmann, Lindner and Rodger to reduce the spectrum problem for odd cycle systems of the complete graph, we show that if there is a $k$-sun system of $K_v$ ($k$ odd) whenever $v$ lies

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 = n p + 1 points, where p is prime and 2 ⩽ n ⩽ 22 . We develop a procedure to obtain exceptions for such type‐1

A partial difference set $S$ in a finite group $G$ satisfying $1 \notin S$ and $S = S^{-1}$ corresponds to an undirected Cayley graph ${\rm Cay}(G,S)$. While the case when $G$ is abelian has been thoroughly studied, there are comparatively few results when $G$ is nonabelian. In this paper, we provide restrictions on the parameters of a partial difference set that apply to both abelian and nonabelian

Square Heffter arrays are $n\times n$ arrays such that each row and each column contains $k$ filled cells, each row and column sum is divisible by $2nk+1$ and either $x$ or $-x$ appears in the array for each integer $1\leq x\leq nk$. Archdeacon noted that a Heffter array, satisfying two additional conditions, yields a face $2$-colourable embedding of the complete graph $K_{2nk+1}$ on an orientable

Many deep, mysterious connections have been observed between collections of mutually unbiased bases (MUBs) and combinatorial designs called $k$-nets (and in particular, between complete collections of MUBs and finite affine - or equivalently: finite projective - planes). Here we introduce the notion of a $k$-net over an algebra $\mathfrak{A}$ and thus provide a common framework for both objects. In

We complete the existence spectrum of perfect Mendelsohn designs PMD (v, 5 ) as v ≡ 0, 1 (mod 5), v ≠ 6, 10 by exhibiting previously unknown designs PMD (15, 5) and PMD (20, 5).

By Raaphorst et al, for a prime power q , covering arrays (CAs) with strength 3 and index 1, defined over the alphabet F q , were constructed using the output of linear feedback shift registers defined by cubic primitive polynomials in F q [ x ] . These arrays have 2 q 3 − 1 rows and q 2 + q + 1 columns. We generalize this construction to apply to all polynomials; provide a new proof that CAs are indeed

This paper introduces almost partitionable sets to generalize the known concept of partitionable sets. These notions provide a unified frame to construct $\mathbb{Z}$-cyclic patterned starter whist tournaments and cyclic balanced sampling plans excluding contiguous units. The existences of partitionable sets and almost partitionable sets are investigated. As an application, a large number of maximum

We investigate how Legendre $G$-array pairs are related to several different perfect binary $G$-array families. In particular we study the relations between Legendre $G$-array pairs, Sidelnikov-Lempel-Cohn-Eastman $\mathbb{Z}_{q-1}$-arrays, Yamada-Pott $G$-array pairs, Ding-Helleseth-Martinsen $\mathbb{Z}_{2}\times \mathbb{Z}_p^{m}$-arrays, Yamada $\mathbb{Z}_{(q-1)/2}$-arrays, Szekeres $\mathbb{Z}^m_{p}$-array

It is well known to us that a graph of diameter l has at least l + 1 eigenvalues. A graph is said to be Laplacian (resp, normalized Laplacian) l ‐extremal if it is of diameter l having exactly l + 1 distinct Laplacian (resp, normalized Laplacian) eigenvalues. A graph is split if its vertex set can be partitioned into a clique and a stable set. Each split graph is of diameter at most 3. In this paper

We define a Mendelsohn triple system (MTS) of order coprime with 3, and having multiplication affine over an abelian group, to be affine, nonramified. By exhibiting a one‐to‐one correspondence between isomorphism classes of affine MTS and those of modules over the Eisenstein integers, we solve the isomorphism problem for affine, nonramified MTS and enumerate these isomorphism classes (extending the

Let C 4 be a cycle of order 4. Write e x ( n , n , n , C 4 ) for the maximum number of edges in a balanced 3‐partite graph whose vertex set consists of three parts, each has n vertices that have no subgraph isomorphic to C 4 . In this paper, we show that e x ( n , n , n , C 4 ) ≥ 3 2 n ( p + 1 ) , where n = p ( p − 1 ) 2 and p is a prime number. Note that e x ( n , n , n , C 4 ) ≤ ( 3 2 2 + o ( 1 )

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.

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

It is known that in any $r$-coloring of the edges of a complete $r$-uniform hypergraph, there exists a spanning monochromatic component. Given a Steiner triple system on $n$ vertices, what is the largest monochromatic component one can guarantee in an arbitrary 3-coloring of the edges? Gyarfas proved that $(2n+3)/3$ is an absolute lower bound and that this lower bound is best possible for infinitely

In this paper we relate t-designs to a forbidden configuration problem in extremal set theory. Let 1_t 0_l denote a column of t 1's on top of l 0's. We assume t>l. Let q. (1_t 0_l) denote the (t+l)xq matrix consisting of t rows of q 1's and l rows of q 0's. We consider extremal problems for matrices avoiding certain submatrices. Let A be a (0,1)-matrix forbidding any (t+l)x(\lambda+2) submatrix (\lambda+2)

The maximum independence number of Steiner triple systems of order v is well‐known. Motivated by questions of access balancing in storage systems, we determine the maximum total cardinality of a pair of disjoint independent sets of Steiner triple systems of order v for all admissible orders.

Let X be a finite set with v elements, called points and β be a family of subsets of X , called blocks. A pair ( X , β ) is called λ ‐design whenever ∣ β ∣ = ∣ X ∣ and 1. for all B i , B j ∈ β , i ≠ j , ∣ B i ∩ B j ∣ = λ ; 2. for all B j ∈ β , ∣ B j ∣ = k j > λ , and not all k j are equal. The only known examples of λ ‐designs are so‐called type‐1 designs, which are obtained from symmetric designs

Group divisible designs (GDDs) with block size 4 and at most 30 points are known for all feasible group types except three, namely 2 3 5 4 , 3 5 6 2 , and 2 2 5 5 . In this paper we provide solutions for the first two of these three 4‐GDDs without assuming any automorphisms. We also construct several other 4‐GDDs. These include classes of 4‐GDDs of types ( 3 m ) 4 ( 6 m ) q ( 3 n ) 1 for 0 ≤ n ≤ (

Deep learning is a machine learning methodology using multi-layer neural network. A multi-layer neural network can be regarded as a chain of complete bipartite graphs. The nodes of the first partita is the input layer and the last is the output layer. The edges of a bipartite graph function as weights which are represented as a matrix. The values of i-th partita are computed by multiplication of the

A Deza graph with parameters $(v,k,b,a)$ is a $k$-regular graph on $v$ vertices in which the number of common neighbors of two distinct vertices takes two values $a$ or $b$ ($a\leq b$) and both cases exist. In the previous papers Deza graphs with parameters $(v,k,b,a)$ where $k-b = 1$ were characterized. In the paper we characterise Deza graphs with $k-b = 2$.

For positive integers n ≥ k ≥ t , a collection B of k ‐subsets of an n ‐set X is called a t ‐packing if every t ‐subset of X appears in at most one set in B . In this paper, we investigate the existence of the maximum 3‐packings whenever n is sufficiently larger than k . When n ≢ 2 ( mod k − 2 ) , the optimal value for the size of a 3‐packing is settled. In other cases, lower and upper bounds are obtained

A pair of orthogonal latin cubes of order $q$ is equivalent to an MDS code with distance $3$ or to an ${\rm OA}_1(3,5,q)$ orthogonal array. We construct pairs of orthogonal latin cubes for a sequence of previously unknown orders $q_i=16(18i-1)+4$ and $q'_i=16(18i+5)+4$. The minimal new obtained parameters of orthogonal arrays are ${\rm OA}_1(3,5,84)$. Keywords: latin square, latin cube, MOLS, MDS code

Several varieties of quasigroups obtained from perfect Mendelsohn designs with block size 4 are defined. One of these is obtained from the so‐called directed standard construction and satisfies the law xy ⋅ (y ⋅ xy) = x and another satisfies Stein's third law xy ⋅ yx = y. Such quasigroups which satisfy the flexible law x ⋅ yx = xy ⋅ x are investigated and characterized. Quasigroups which satisfy both