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Intercalates in double and triple arrays J. Comb. Des. (IF 0.667) Pub Date : 20211124
Tomas NilsonThis paper addresses the question of how intercalates occur in the two known infinite families of triple arrays, the Paley triple arrays constructed in 2005 by Preece et al., and the Triple arrays from difference sets in 2017 by Nilson and Cameron. The main reason for doing this is that the number of such embedded Latin squares is often used when checking whether two arrays are isotopic or not. We

Reconfiguration of subspace partitions J. Comb. Des. (IF 0.667) Pub Date : 20210930
Fusun Akman, Papa A. SissokhoLet q be a fixed prime power and let V ( n , q ) denote a vector space of dimension n over the Galois field with q elements. A subspace partition (also called “vector space partition”) of V ( n , q ) is a collection of subspaces of V ( n , q ) with the property that every nonzero element of V ( n , q ) appears in exactly one of these subspaces. Given positive integers a , b , n such that 1 ≤ a < b

The combinatorial game Nofil played on Steiner triple systems J. Comb. Des. (IF 0.667) Pub Date : 20211018
Melissa A. Huggan, Svenja Huntemann, Brett StevensWe introduce an impartial combinatorial game on Steiner triple systems called Next One to Fill Is the Loser (Nofil). Players move alternately, choosing points of the triple system. If a player is forced to fill a block on their turn, they lose. By computing nimvalues, we determine optimal strategies for Nofil on all Steiner triple systems up to order 15 and a sampling for orders 19, 21 and 25. The

Generalized pentagonal geometries J. Comb. Des. (IF 0.667) Pub Date : 20211031
A. D. Forbes, C. G. RutherfordA pentagonal geometry PENT( k , r) is a partial linear space, where every line 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. Here we generalize the concept by allowing the points not collinear with x to form the point set of a Steiner system S ( 2 , k , w ) whose blocks are lines

Moufang sets generated by translations in unitals J. Comb. Des. (IF 0.667) Pub Date : 20211116
Theo Grundhöfer, Markus J. Stroppel, Hendrik Van MaldeghemWe consider unitals of order q with two points which are centers of translation groups of order q. The group G generated by these translations induces a Moufang set on the block joining the two points. We show that G is either SL ( 2 , F q ) (as in all classical unitals and also in some nonclassical examples), or PSL ( 2 , F q ) , or a Suzuki, or a Ree group. Moreover, G is semiregular outside the

On optimal ( Z 6 m × Z 6 n , 4 , 1 ) and ( Z 2 m × Z 18 n , 4 , 1 ) difference packings and their related codes J. Comb. Des. (IF 0.667) Pub Date : 20211116
Jingyuan Chen, Lijun JiWe give a direct construction of a ( Z p × G , { 0 } × G , 4 , 1 ) relative difference family for G ∈ { Z 6 × Z 6 , Z 2 × Z 18 , Z 6 × Z 18 , Z 2 × Z 54 } and every prime p ≡ 3 ( mod 4 ) with p > 3. These allow us to construct an optimal ( Z 6 m × Z 6 n , 4 , 1 ) difference packing and an optimal ( Z 2 m × Z 18 n , 4 , 1 ) difference packing for every pair of positive integers ( m , n ) . The corresponding

Enumeration of Latin squares with conjugate symmetry J. Comb. Des. (IF 0.667) Pub Date : 20211116
Brendan D. McKay, Ian M. WanlessA Latin square has six conjugate Latin squares obtained by uniformly permuting its (row, column, symbol) triples. We say that a Latin square has conjugate symmetry if at least two of its six conjugates are equal. We enumerate Latin squares with conjugate symmetry and classify them according to several common notions of equivalence. We also do similar enumerations under additional hypotheses, such as

A note on tight projective 2designs J. Comb. Des. (IF 0.667) Pub Date : 20210905
Joseph W. Iverson, Emily J. King, Dustin G. MixonWe study tight projective 2designs in three different settings. In the complex setting, Zauner's conjecture predicts the existence of a tight projective 2design in every dimension. Pandey, Paulsen, Prakash, and Rahaman recently proposed an approach to make quantitative progress on this conjecture in terms of the entanglement breaking rank of a certain quantum channel. We show that this quantity is

Extended near Skolem sequences Part II J. Comb. Des. (IF 0.667) Pub Date : 20210915
Catharine A. Baker, Vaclav Linek, Nabil ShalabyA kextended qnear 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 ) , where ( n , q , k ) ≠ (

Legendre pairs of lengths ℓ ≡ 0 ( mod 3 ) J. Comb. Des. (IF 0.667) Pub Date : 20210930
Ilias Kotsireas, Christoph KoutschanWe prove a proposition that connects constantperiodic autocorrelation function sequences and the corresponding Legendre pairs with integer power spectral density values. We show how to determine explicitly the complete spectrum of the ( ℓ ∕ 3 ) rd value of the discrete Fourier transform for Legendre pairs of lengths ℓ ≡ 0 ( mod 3 ) . This is accomplished by two new algorithms based on numbertheoretic

Maximum wcyclic holey group divisible packings with block size three and applications to optical orthogonal codes J. Comb. Des. (IF 0.667) Pub Date : 20210802
Zenghui Fang, Junling Zhou, Lidong WangIn this paper we investigate combinatorial constructions for wcyclic holey group divisible packings with block size three (3HGDPs). For any positive integers u , v , w with u ≡ 0 , 1 ( mod 3 ) , the exact number of base blocks of a maximum wcyclic 3HGDP of type ( u , w v ) is determined. This result is used to determine the exact number of codewords in a maximum threedimensional ( u × v × w

Rowcolumn factorial designs with multiple levels J. Comb. Des. (IF 0.667) Pub Date : 20210720
Fahim Rahim, Nicholas J. CavenaghAn m × n rowcolumn 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 [

Extended near Skolem sequences Part I J. Comb. Des. (IF 0.667) Pub Date : 20210802
Cathy A. Baker, Vaclav Linek, Nabil ShalabyA kextended qnear 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

Perfect and nearly perfect separation dimension of complete and random graphs J. Comb. Des. (IF 0.667) Pub Date : 20210830
Raphael YusterThe separation dimension of a hypergraph G is the smallest natural number d for which there is an embedding of G into R d , such that any pair of disjoint edges is separated by some hyperplane normal to one of the axes. The perfect separation dimension further requires that any pair of disjoint edges is separated by the same amount of such (pairwise nonparallel) hyperplanes. While it is known that

Resolvable cycle decompositions of complete multigraphs and complete equipartite multigraphs via layering and detachment J. Comb. Des. (IF 0.667) Pub Date : 20210622
Amin Bahmanian, Mateja ŠajnaWe 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 2factorizations of complete multigraphs from existing 2factorizations of complete graphs, and detachment, which allows us to construct resolvable

Infinite families of 2designs derived from affineinvariant codes J. Comb. Des. (IF 0.667) Pub Date : 20210622
Yan Liu, Xiwang CaoIn this paper, we first study a general type of affineinvariant codes and their support 2designs. Then we derive infinite families of 2designs from a special class of affineinvariant codes and obtain their parameters by determining the weight distribution of the linear codes.

On strongly regular designs admitting fusion to strongly regular decomposition J. Comb. Des. (IF 0.667) Pub Date : 20210720
A. D. SankeyA 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

More constructions for Sperner partition systems J. Comb. Des. (IF 0.667) Pub Date : 20210604
Adam Gowty, Daniel HorsleyAn ( 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

Universal pary designs J. Comb. Des. (IF 0.667) Pub Date : 20210529
Liam JolliffeWe 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 nonnull designs and determine necessary and sufficient conditions on the coefficients for such a design to exist. This allows us to classify all universal designs

On planar arcs of size ( q + 3 ) ∕ 2 J. Comb. Des. (IF 0.667) Pub Date : 20210606
Gülizar Günay, Michel LavrauwThe 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

Reduction for flagtransitive 2 ( v , k , λ ) designs with ( r , λ ) = 2 J. Comb. Des. (IF 0.667) Pub Date : 20210615
Yanwei Zhao, Shenglin ZhouThis paper studies flagtransitive 2 ( v , k , λ ) designs with ( r , λ ) = 2 . We prove that if D is a flagtransitive nontrivial 2 ( v , k , λ ) design with ( r , λ ) = 2 , then G is of affine or almost simple type. Furthermore, we analyze the flagtransitive 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.

Small Latin arrays have a near transversal J. Comb. Des. (IF 0.667) Pub Date : 20210504
Darcy Best, Kyle Pula, Ian M. WanlessA 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 extended binary quadratic residue code of length 42 holds a 3design J. Comb. Des. (IF 0.667) Pub Date : 20210503
A. Bonnecaze, P. SoléThe 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.

Evasive subspaces J. Comb. Des. (IF 0.667) Pub Date : 20210524
Daniele Bartoli, Bence Csajbók, Giuseppe Marino, Rocco TrombettiLet 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

On flagtransitive imprimitive 2designs J. Comb. Des. (IF 0.667) Pub Date : 20210524
Alice Devillers, Cheryl E. PraegerIn 1987, Huw Davies proved that, for a flagtransitive pointimprimitive 2 ( v , k , λ ) design, both the blocksize 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

Further results on large sets of partitioned incomplete Latin squares J. Comb. Des. (IF 0.667) Pub Date : 20210321
Cong Shen, Dongliang Li, Haitao CaoIn 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

Explicit Baranyai partitions for quadruples, Part I: Quadrupling constructions J. Comb. Des. (IF 0.667) Pub Date : 20210412
Yeow Meng Chee, Tuvi Etzion, Han Mao Kiah, Alexander Vardy, Chengmin WangIt 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

Maximal partial Room squares J. Comb. Des. (IF 0.667) Pub Date : 20210419
Mariusz Meszka, Alexander RosaA 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 note on magic rectangle set MRSΓ(2k+1,4;4l+2) J. Comb. Des. (IF 0.667) Pub Date : 20210502
Sylwia Cichacz, Tomasz HincA 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 }

Perfect 2‐colorings of Hamming graphs J. Comb. Des. (IF 0.667) Pub Date : 20210310
Evgeny A. Bespalov, Denis S. Krotov, Aleksandr A. Matiushev, Anna A. Taranenko, Konstantin V. Vorob'evWe 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

Colouring problems for symmetric configurations with block size 3 J. Comb. Des. (IF 0.667) Pub Date : 20210321
Grahame Erskine, Terry Griggs, Jozef ŠiráňThe 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

Corrigendum: Super‐simple resolvable balanced incomplete block designs with block size 4 and index 2 J. Comb. Des. (IF 0.667) Pub Date : 20210309
Xiande Zhang, Gennian GeIn 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.

Partial geometric designs with block sizes three and four J. Comb. Des. (IF 0.667) Pub Date : 20210119
Jing Qu, Jianguo Lei, Xiuling ShanIn 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

Pentagonal geometries with block sizes 3, 4, and 5 J. Comb. Des. (IF 0.667) Pub Date : 20210111
Anthony D. ForbesA 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

Latin squares from multiplication tables J. Comb. Des. (IF 0.667) Pub Date : 20210125
Michał Dębski, Jarosław GrytczukConsider 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

Large families of permutations of Z n whose pairwise sums are permutations J. Comb. Des. (IF 0.667) Pub Date : 20201222
Bojan Bašić, Stefan HačkoWe 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

Existence of strong difference families and constructions for eight new 2‐designs J. Comb. Des. (IF 0.667) Pub Date : 20210104
Xiaomiao Wang, Tao Feng, Shixin WangUsing 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

The existence of r‐golf designs J. Comb. Des. (IF 0.667) Pub Date : 20210112
Xiangqian Li, Yanxun Chang, Junling ZhouAn 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

Block‐avoiding point sequencings J. Comb. Des. (IF 0.667) Pub Date : 20210202
Simon R. Blackburn, Tuvi EtzionLet 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

Quaternary complex Hadamard matrices of order 18 J. Comb. Des. (IF 0.667) Pub Date : 20201125
Patric R. J. Östergård, William T. PaavolaA 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

Sparse Steiner triple systems of order 21 J. Comb. Des. (IF 0.667) Pub Date : 20201117
Janne I. Kokkala, Patric R. J. ÖstergårdA ( 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

Solution to the outstanding case of the spouse‐loving variant of the Oberwolfach problem with uniform cycle length J. Comb. Des. (IF 0.667) Pub Date : 20201110
Andiyappan Shanmuga Vadivu, Lakshmanan Panneerselvam, Appu MuthusamyLet 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

Intersecting and 2‐intersecting hypergraphs with maximal covering number: The Erdős–Lovász theme revisited J. Comb. Des. (IF 0.667) Pub Date : 20201208
János BarátErdős and Lovasz noticed that an $r$uniform intersecting hypergraph $H$ with maximal covering number, that is $\tau(H)=r$, must have at least $\frac{8}{3}r3$ 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

The localization number of designs J. Comb. Des. (IF 0.667) Pub Date : 20201207
Anthony Bonato, Melissa A. Huggan, Trent G. MarbachWe 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

Graph decompositions in projective geometries J. Comb. Des. (IF 0.667) Pub Date : 20201201
Marco Buratti, Anamari Nakić, Alfred WassermannLet PG$(\mathbb{F}_q^v)$ be the $(v1)$dimensional projective space over $\mathbb{F}_q$ and let $\Gamma$ be a simple graph of order ${q^k1\over q1}$ 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 vertexset of every block is a subspace of PG$(\mathbb{F}_q^v)$; every two distinct

Anti‐Pasch optimal coverings with triples J. Comb. Des. (IF 0.667) Pub Date : 20201125
Fatih Demirkale, Diane Donovan, Mike GrannellIt 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.

Pure tetrahedral quadruple systems with index two J. Comb. Des. (IF 0.667) Pub Date : 20201023
Ruijing Liu, Junling ZhouAn 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 reduction of the spectrum problem for odd sun systems and the prime case J. Comb. Des. (IF 0.667) Pub Date : 20201030
Marco Buratti, Anita Pasotti, Tommaso TraettaA $k$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

A note on λ‐designs J. Comb. Des. (IF 0.667) Pub Date : 20200928
Ajeet K. Yadav, Rajendra M. Pawale, Mohan S. ShrikhandeLet 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

Restrictions on parameters of partial difference sets in nonabelian groups J. Comb. Des. (IF 0.667) Pub Date : 20201007
Eric Swartz, Gabrielle TauscheckA 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

Biembeddings of cycle systems using integer Heffter arrays J. Comb. Des. (IF 0.667) Pub Date : 20200928
Nicholas J. Cavenagh, Diane M. Donovan, Emine Ş. YazıcıSquare 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

Rigidity and a common framework for mutually unbiased bases and k‐nets J. Comb. Des. (IF 0.667) Pub Date : 20200825
Sloan Nietert, Zsombor Szilágyi, Mihály WeinerMany 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

The last two perfect Mendelsohn designs with block size 5 J. Comb. Des. (IF 0.667) Pub Date : 20200818
Terry S. Griggs, Andrew R. KozlikWe 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).

An extension of a construction of covering arrays J. Comb. Des. (IF 0.667) Pub Date : 20200804
Daniel Panario, Mark Saaltink, Brett Stevens, Daniel WevrickBy 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

Partitionable sets, almost partitionable sets, and their applications J. Comb. Des. (IF 0.667) Pub Date : 20200720
Yanxun Chang, Simone Costa, Tao Feng, Xiaomiao WangThis 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

Legendre G‐array pairs and the theoretical unification of several G‐array families J. Comb. Des. (IF 0.667) Pub Date : 20200715
K. T. Arasu, D. A. Bulutoglu, J. R. HollonWe 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, SidelnikovLempelCohnEastman $\mathbb{Z}_{q1}$arrays, YamadaPott $G$array pairs, DingHellesethMartinsen $\mathbb{Z}_{2}\times \mathbb{Z}_p^{m}$arrays, Yamada $\mathbb{Z}_{(q1)/2}$arrays, Szekeres $\mathbb{Z}^m_{p}$array

On split graphs with three or four distinct (normalized) Laplacian eigenvalues J. Comb. Des. (IF 0.667) Pub Date : 20200713
Shuchao Li, Wanting SunIt 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

Affine Mendelsohn triple systems and the Eisenstein integers J. Comb. Des. (IF 0.667) Pub Date : 20200624
Alex W. NowakWe 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

A note on 3‐partite graphs without 4‐cycles J. Comb. Des. (IF 0.667) Pub Date : 20200622
Zequn Lv, Mei Lu, Chunqiu FangLet 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 )

Ternary codes, biplanes, and the nonexistence of some quasisymmetric and quasi3 designs J. Comb. Des. (IF 0.667) Pub Date : 20200622
Akihiro Munemasa, Vladimir D. TonchevThe 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$(267,57,12)$ quasi3 design. The nonexistence of a 2$(149,37,9)$ quasi3 design is also proved.