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Alternating parity weak sequencing J. Comb. Des. (IF 0.7) Pub Date : 2024-02-27 Simone Costa, Stefano Della Fiore
A subset of a group is ‐weakly sequenceable if there is an ordering of its elements such that the partial sums , given by and for , satisfy whenever and . By Costa et al., it was proved that if the order of a group is then all sufficiently large subsets of the nonidentity elements are ‐weakly sequenceable when is prime, and . Inspired by this result, we show that, if is the semidirect product of and
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The existence of 23 ${2}^{3}$‐decomposable super‐simple (v,4,6) $(v,4,6)$‐BIBDs J. Comb. Des. (IF 0.7) Pub Date : 2024-02-23 Huangsheng Yu, Jingyuan Chen, R. Julian R. Abel, Dianhua Wu
A design is said to be super‐simple if the intersection of any two blocks has at most two elements. A design with index is said to be ‐decomposable, if its blocks can be partitioned into nonempty collections , , such that each with the point set forms a design with index . In this paper, it is proved that there exists a ‐decomposable super‐simple ‐BIBD (balanced incomplete block design) if and only
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Partition of ordered triples into uniform holey ordered designs J. Comb. Des. (IF 0.7) Pub Date : 2024-02-19 Yuli Tan, Junling Zhou
A large set is a partition of all ordered triples of a ‐set into disjoint ordered designs of order . In this paper, we generalize the large set with to the notion of , representing a partition of all ordered triples of a ‐set into disjoint uniform holely ordered designs s. We show that a exists if and only if and , except for . Moreover, we study the existence of a with every member having a kind of
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Geometric dual and sum-rank minimal codes J. Comb. Des. (IF 0.7) Pub Date : 2024-02-14 Martino Borello, Ferdinando Zullo
The main purpose of this paper is to further study the structure, parameters and constructions of the recently introduced minimal codes in the sum-rank metric. These objects form a bridge between the classical minimal codes in the Hamming metric, the subject of intense research over the past three decades partly because of their cryptographic properties, and the more recent rank-metric minimal codes
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Constructing MRD codes by switching J. Comb. Des. (IF 0.7) Pub Date : 2024-02-08 Minjia Shi, Denis S. Krotov, Ferruh Özbudak
Maximum rank-distance (MRD) codes are (not necessarily linear) maximum codes in the rank-distance metric space on m$m$-by-n$n$ matrices over a finite field Fq${{\mathbb{F}}}_{q}$. They are diameter perfect and have the cardinality qm(n−d+1)${q}^{m(n-d+1)}$ if m≥n$m\ge n$. We define switching in MRD codes as the replacement of special MRD subcodes by other subcodes with the same parameters. We consider
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New constructions for disjoint partial difference families and external partial difference families J. Comb. Des. (IF 0.7) Pub Date : 2024-02-04 Sophie Huczynska, Laura Johnson
Recently, new combinatorial structures called disjoint partial difference families (DPDFs) and external partial difference families (EPDFs) were introduced, which simultaneously generalize partial difference sets, disjoint difference families and external difference families, and have applications in information security. So far, all known construction methods have used cyclotomy in finite fields.
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Sailing league problems J. Comb. Des. (IF 0.7) Pub Date : 2024-01-22 Robert Schüler, Achill Schürmann
We describe a class of combinatorial design problems which typically occur in professional sailing league competitions. We discuss connections to resolvable block designs and equitable coverings and to scheduling problems in operations research. We in particular give suitable boolean quadratic and integer linear optimization problem formulations, as well as further heuristics and restrictions, that
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Sporadic simple groups as flag-transitive automorphism groups of symmetric designs J. Comb. Des. (IF 0.7) Pub Date : 2023-12-20 Seyed Hassan Alavi, Ashraf Daneshkhah
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
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Reduction for flag-transitive point-primitive 2-(v, k, λ) designs with λ prime J. Comb. Des. (IF 0.7) Pub Date : 2023-11-28 Yongli Zhang, Jianfu Chen
It is shown that the flag-transitive, point-primitive automorphism groups of 2- ( v , k , λ ) $(v,k,\lambda )$ designs with λ $\lambda $ prime must be of affine type or almost simple type.
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Incidence-free sets and edge domination in incidence graphs J. Comb. Des. (IF 0.7) Pub Date : 2023-11-23 Sam Spiro, Sam Adriaensen, Sam Mattheus
A set of edges Γ ${\rm{\Gamma }}$ of a graph G $G$ is an edge dominating set if every edge of G $G$ intersects at least one edge of Γ ${\rm{\Gamma }}$ , and the edge domination number γ e ( G ) ${\gamma }_{e}(G)$ is the smallest size of an edge dominating set. Expanding on work of Laskar and Wallis, we study γ e ( G ) ${\gamma }_{e}(G)$ for graphs G $G$ which are the incidence graph of some incidence
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On symmetric designs with flag-transitive and point-quasiprimitive automorphism groups J. Comb. Des. (IF 0.7) Pub Date : 2023-11-05 Zhilin Zhang, Jianfu Chen, Shenglin Zhou
Let D = ( P , ℬ ) be a nontrivial symmetric ( v , k , λ ) -design with λ ≤ 100 , and let G be a flag-transitive automorphism group of D . In this paper, we show that if G is quasiprimitive on P , then G is of holomorph affine or almost simple type. Moreover, if G is imprimitive on P , then G is of almost simple type. According to this observation and to the classification of the finite simple groups we
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Dual incidences and t-designs in vector spaces J. Comb. Des. (IF 0.7) Pub Date : 2023-10-13 Kristijan Tabak
Let V $V$ be an n $n$ -dimensional vector space over F q ${{\mathbb{F}}}_{q}$ and H ${\rm{ {\mathcal H} }}$ is any set of k $k$ -dimensional subspaces of V $V$ . We construct two incidence structures D m a x ( H ) ${{\mathscr{D}}}_{max}({\rm{ {\mathcal H} }})$ and D m i n ( H ) ${{\mathscr{D}}}_{min}({\rm{ {\mathcal H} }})$ using subspaces from H ${\rm{ {\mathcal H} }}$ . The points are subspaces from
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Completing the solution of the directed Oberwolfach problem with cycles of equal length J. Comb. Des. (IF 0.7) Pub Date : 2023-09-25 Alice Lacaze-Masmonteil
In this paper, we give a solution to the last outstanding case of the directed Oberwolfach problem with tables of uniform length. Namely, we address the two-table case with tables of equal odd length. We prove that the complete symmetric digraph on 2 m $2m$ vertices, denoted K 2 m * ${K}_{2m}^{* }$ , admits a resolvable decomposition into directed cycles of odd length m $m$ . This completely settles
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Orthogonal cycle systems with cycle length less than 10 J. Comb. Des. (IF 0.7) Pub Date : 2023-09-25 Selda Küçükçifçi, Emine Şule Yazıcı
An H $H$ -decomposition of a graph G $G$ is a partition of the edge set of G $G$ into subsets, where each subset induces a copy of the graph H $H$ . A k $k$ -orthogonal H $H$ -decomposition of G $G$ is a set of k $k$ H $H$ -decompositions of G $G$ such that any two copies of H $H$ in distinct H $H$ -decompositions intersect in at most one edge. When G = K v $G={K}_{v}$ , we call the H $H$ -decomposition
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Generalizations of some Nordhaus–Gaddum-type results on spectral radius J. Comb. Des. (IF 0.7) Pub Date : 2023-09-21 Junying Lu, Lanchao Wang, Yaojun Chen
Let G $G$ be a simple graph and λ ( G ) $\lambda (G)$ the spectral radius of G $G$ . For k ≥ 2 $k\ge 2$ , a k $k$ -edge decomposition ( H 1 , … , H k ) $({H}_{1},{\rm{\ldots }},{H}_{k})$ is k $k$ spanning subgraphs such that their edge sets form a k $k$ -partition of the edge set of G $G$ . In this paper, we obtain some sharp lower and upper bounds for λ ( H 1 ) + ⋯ + λ ( H k ) $\lambda ({H}_{1})+\
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Construction of rank 4 self-dual association schemes inducing three partial geometric designs J. Comb. Des. (IF 0.7) Pub Date : 2023-09-05 Akihide Hanaki
Xu characterized rank 4 self-dual association schemes inducing three partial geometric designs by their character tables. We construct such association schemes as Schur rings over Abelian 2-groups.
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proper partial geometries with an automorphism group acting primitively on points and lines J. Comb. Des. (IF 0.7) Pub Date : 2023-08-31 Wendi Di
Let S ${\mathscr{S}}$ be a finite proper partial geometry pg ( s , t , α ) $(s,t,\alpha )$ not isomorphic to the van Lint–Schrijver partial geometry pg ( 5 , 5 , 2 ) $(5,5,2)$ and let G $G$ be a group of automorphisms of S ${\mathscr{S}}$ acting primitively on both points and lines of S ${\mathscr{S}}$ , we show that if α ≤ 60 $\alpha \le 60$ then G $G$ must be almost simple.
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P℘N functions, complete mappings and quasigroup difference sets J. Comb. Des. (IF 0.7) Pub Date : 2023-08-23 Nurdagül Anbar, Tekgül Kalaycı, Wilfried Meidl, Constanza Riera, Pantelimon Stănică
We investigate pairs of permutations F , G $F,G$ of F p n ${{\mathbb{F}}}_{{p}^{n}}$ such that F ( x + a ) − G ( x ) $F(x+a)-G(x)$ is a permutation for every a ∈ F p n $a\in {{\mathbb{F}}}_{{p}^{n}}$ . We show that, in that case, necessarily G ( x ) = ℘ ( F ( x ) ) $G(x)=\wp (F(x))$ for some complete mapping − ℘ $-\wp $ of F p n ${{\mathbb{F}}}_{{p}^{n}}$ , and call the permutation F $F$ a perfect
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On the directed Oberwolfach problem for complete symmetric equipartite digraphs and uniform-length cycles J. Comb. Des. (IF 0.7) Pub Date : 2023-08-22 Nevena Francetić, Mateja Šajna
We examine the necessary and sufficient conditions for a complete symmetric equipartite digraph K n [ m ] * ${K}_{n[m]}^{* }$ with n $n$ parts of size m $m$ to admit a resolvable decomposition into directed cycles of length t $t$ . We show that the obvious necessary conditions are sufficient for m , n , t ≥ 2 $m,n,t\ge 2$ in each of the following four cases: (i) m ( n − 1 ) $m(n-1)$ is even; (ii) gcd
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A generalization of group divisible t $t$-designs J. Comb. Des. (IF 0.7) Pub Date : 2023-08-10 Sijia Liu, Yue Han, Lijun Ma, Lidong Wang, Zihong Tian
Cameron defined the concept of generalized t $t$ -designs, which generalized t $t$ -designs, resolvable designs and orthogonal arrays. This paper introduces a new class of combinatorial designs which simultaneously provide a generalization of both generalized t $t$ -designs and group divisible t $t$ -designs. In certain cases, we derive necessary conditions for the existence of generalized group divisible
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Cameron–Liebler sets for maximal totally isotropic flats in classical affine spaces J. Comb. Des. (IF 0.7) Pub Date : 2023-07-19 Jun Guo, Lingyu Wan
Let A C G ( 2 ν , F q ) $ACG(2\nu ,{{\mathbb{F}}}_{q})$ be the 2 ν $2\nu $ -dimensional classical affine space with parameter e $e$ over a q $q$ -element finite field F q ${{\mathbb{F}}}_{q}$ , and O ν ${{\mathscr{O}}}_{\nu }$ be the set of all maximal totally isotropic flats in A C G ( 2 ν , F q ) $ACG(2\nu ,{{\mathbb{F}}}_{q})$ . In this paper, we discuss Cameron–Liebler sets in O ν ${{\mathscr{O}}}_{\nu
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Exceptional designs in some extended quadratic residue codes J. Comb. Des. (IF 0.7) Pub Date : 2023-07-18 Reina Ishikawa
In the present paper, we give proofs of the existence of a 3-design in the extended ternary quadratic residue code of length 14 and the extended quaternary quadratic residue code of length 18.
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Cycles of quadratic Latin squares and antiperfect 1-factorisations J. Comb. Des. (IF 0.7) Pub Date : 2023-07-10 Jack Allsop
A Latin square of order n $n$ is an n × n $n\times n$ matrix of n $n$ symbols, such that each symbol occurs exactly once in each row and column. For an odd prime power q $q$ let F q ${{\mathbb{F}}}_{q}$ denote the finite field of order q $q$ . A quadratic Latin square is a Latin square L [ a , b ] ${\rm{ {\mathcal L} }}[a,b]$ defined by
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Totally symmetric quasigroups of order 16 J. Comb. Des. (IF 0.7) Pub Date : 2023-07-13 Hy Ginsberg
We present the number of totally symmetric quasigroups (equivalently, totally symmetric Latin squares) of order 16, as well as the number of isomorphism classes of such objects. Totally symmetric quasigroups of orders up to and including 16 that are (respectively) medial, idempotent, and unipotent are also enumerated.
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Enumerating Steiner triple systems J. Comb. Des. (IF 0.7) Pub Date : 2023-07-13 Daniel Heinlein, Patric R. J. Östergård
Steiner triple systems (STSs) have been classified up to order 19. Earlier estimations of the number of isomorphism classes of STSs of order 21, the smallest open case, are discouraging as for classification, so it is natural to focus on the easier problem of merely counting the isomorphism classes. Computational approaches for counting STSs are here considered and lead to an algorithm that is used
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On the existence of k $k$-cycle semiframes for even k $k$ J. Comb. Des. (IF 0.7) Pub Date : 2023-07-13 Li Wang, Haibo Ji, Haitao Cao
A C k ${C}_{k}$ -semiframe of type g u ${g}^{u}$ is a C k ${C}_{k}$ -group divisible design of type g u ( X , G , ℬ ) ${g}^{u}({\mathscr{X}},{\mathscr{G}},{\rm{ {\mathcal B} }})$ in which X ${\mathscr{X}}$ is the vertex set, G ${\mathscr{G}}$ is the group set, and the set ℬ ${\rm{ {\mathcal B} }}$ of k $k$ -cycles can be written as a disjoint union ℬ = P ∪ Q ${\rm{ {\mathcal B} }}={\mathscr{P}}\cup
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The chromatic index of finite projective spaces J. Comb. Des. (IF 0.7) Pub Date : 2023-06-30 Lei Xu, Tao Feng
A line coloring of PG ( n , q ) $\text{PG}(n,q)$ , the n $n$ -dimensional projective space over GF ( q ) $(q)$ , is an assignment of colors to all lines of PG ( n , q ) $\text{PG}(n,q)$ so that any two lines with the same color do not intersect. The chromatic index of PG ( n , q ) $\text{PG}(n,q)$ , denoted by χ ′ ( PG ( n , q ) ) $\chi ^{\prime} (\text{PG}(n,q))$ , is the least number of colors for
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Existence of small ordered orthogonal arrays J. Comb. Des. (IF 0.7) Pub Date : 2023-06-19 Kai-Uwe Schmidt, Charlene Weiß
We show that there exist ordered orthogonal arrays, whose sizes deviate from the Rao bound by a factor that is polynomial in the parameters of the ordered orthogonal array. The proof is nonconstructive and based on a probabilistic method due to Kuperberg, Lovett and Peled.
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Linear and circular single-change covering designs revisited J. Comb. Des. (IF 0.7) Pub Date : 2023-06-01 Amanda Chafee, Brett Stevens
A single-change covering design (SCCD) is a v $v$ -set X $X$ and an ordered list ℒ ${\rm{ {\mathcal L} }}$ of b $b$ blocks of size k $k$ where every pair from X $X$ must occur in at least one block. Each pair of consecutive blocks differs by exactly one element. This is a linear single-change covering design, or more simply, a single-change covering design. A single-change covering design is circular
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Self-dual association schemes, fusions of Hamming schemes, and partial geometric designs J. Comb. Des. (IF 0.7) Pub Date : 2023-05-25 Bangteng Xu
Partial geometric designs can be constructed from basic relations of association schemes. An infinite family of partial geometric designs were constructed from the fusion schemes of certain Hamming schemes in work by Nowak et al. (2016). A general method to create partial geometric designs from association schemes is given by Xu (2023). In this paper, we continue the research by Xu (2023). We will
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Primitive C 4 ${C}_{4}$-decompositions of K n − I ${K}_{n}-I$ J. Comb. Des. (IF 0.7) Pub Date : 2023-05-05 Michael W. Schroeder
A decomposition C ${\mathscr{C}}$ of a graph G $G$ is primitive if no proper, nontrivial subset of C ${\mathscr{C}}$ is a decomposition of an induced subgraph of G $G$ . An unresolved question posed by Asplund et al. in a recent publication involves the existence of primitive decompositions of cocktail party graphs into cycles of length 4, which is resolved by this paper.
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Magic partially filled arrays on abelian groups J. Comb. Des. (IF 0.7) Pub Date : 2023-05-04 Fiorenza Morini, Marco Antonio Pellegrini
In this paper we introduce a special class of partially filled arrays. A magic partially filled array MPF Ω ( m , n ; s , k ) ${\text{MPF}}_{{\rm{\Omega }}}(m,n;s,k)$ on a subset Ω ${\rm{\Omega }}$ of an abelian group ( Γ , + ) $({\rm{\Gamma }},+)$ is a partially filled array of size m × n $m\times n$ with entries in Ω ${\rm{\Omega }}$ such that (i) every ω ∈ Ω $\omega \in {\rm{\Omega }}$ appears once
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On an Assmus–Mattson type theorem for type I and even formally self-dual codes J. Comb. Des. (IF 0.7) Pub Date : 2023-04-17 Tsuyoshi Miezaki, Hiroyuki Nakasora
In the present paper, we give an Assmus–Mattson type theorem for near-extremal Type I and even formally self-dual codes. We show the existence of 1-designs or 2-designs for these codes. As a corollary, we prove the uniqueness of a self-orthogonal 2- ( 16 , 6 , 8 ) $(16,6,8)$ design.
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The existence of λ $\lambda $-decomposable super-simple (4,2λ) $(4,2\lambda )$-GDDs of type gu ${g}^{u}$ with λ=2,4 $\lambda =2,4$ J. Comb. Des. (IF 0.7) Pub Date : 2023-03-30 Huangsheng Yu, Jingyuan Chen, R. Julian R. Abel, Dianhua Wu
A design is said to be super-simple if the intersection of any two of its blocks has at most two elements. A design with index t λ $t\lambda $ is said to be λ $\lambda $ -decomposable, if its blocks can be partitioned into nonempty collections ℬ i ${{\rm{ {\mathcal B} }}}_{i}$ , 1 ≤ i ≤ t $1\le i\le t$ , such that each ℬ i ${{\rm{ {\mathcal B} }}}_{i}$ with the point set forms a design with index λ
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Ordered covering arrays and upper bounds on covering codes J. Comb. Des. (IF 0.7) Pub Date : 2023-03-30 André Guerino Castoldi, Emerson L. Monte Carmelo, Lucia Moura, Daniel Panario, Brett Stevens
This work shows several direct and recursive constructions of ordered covering arrays (OCAs) using projection, fusion, column augmentation, derivation, concatenation, and Cartesian product. Upper bounds on covering codes in Niederreiter–Rosenbloom–Tsfasman (shorten by NRT) spaces are also obtained by improving a general upper bound. We explore the connection between ordered covering arrays and covering
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Alternating groups and point-primitive linear spaces with number of points being squarefree J. Comb. Des. (IF 0.7) Pub Date : 2023-02-26 Haiyan Guan, Shenglin Zhou
This paper is a further contribution to the classification of point-primitive finite regular linear spaces. Let S${\mathscr{S}}$ be a nontrivial finite regular linear space whose number of points v$v$ is squarefree. We prove that if ◂...▸G≤Aut(S)$G\le \text{Aut}({\mathscr{S}})$ is point-primitive with an alternating socle, then S${\mathscr{S}}$ is the projective space ◂...▸PG(3,2)$\text{PG}(3,2)$.
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Towards the Ryser–Woodall λ $\lambda $-design conjecture J. Comb. Des. (IF 0.7) Pub Date : 2023-02-26 Navin M. Singhi, Mohan S. Shrikhande, Rajendra M. Pawale
Let r1${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=gcd◂()▸(◂−▸r1−r2gcd◂()▸(r1−1
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On the chromatic number of some generalized Kneser graphs J. Comb. Des. (IF 0.7) Pub Date : 2023-02-10 Jozefien D'haeseleer, Klaus Metsch, Daniel Werner
We determine the chromatic number of the Kneser graph qΓ◂,▸7,{3,4}$q{{\rm{\Gamma }}}_{7,\{3,4\}}$ of flags of vectorial type {3,4}$\{3,4\}$ of a rank 7 vector space over the finite field GF(q)$\mathrm{GF}(q)$ for large q$q$ and describe the colorings that attain the bound. This result relies heavily, not only on the independence number, but also on the structure of all large independent sets. Furthermore
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Balanced covering arrays: A classification of covering arrays and packing arrays via exact methods J. Comb. Des. (IF 0.7) Pub Date : 2023-02-05 Ludwig Kampel, Irene Hiess, Ilias S. Kotsireas, Dimitris E. Simos
In this paper we investigate the intersections of classes of covering arrays (CAs) and packing arrays (PAs). The arrays appearing in these intersections obey to upper and lower bounds regarding the appearance of tuples in sub-matrices—we call these arrays balanced covering arrays. We formulate and formalize first observations for which upper and lower bounds on the appearance of tuples it is of interest
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The maximum number of columns in E(s2) $\,E({s}^{2})$-optimal supersaturated designs with 16 rows and smax=4 ${s}_{{\rm{\max }}}=4$ is 60 J. Comb. Des. (IF 0.7) Pub Date : 2023-01-09 Luis B. Morales
We show that the maximum number of columns in E(s2)$\,E({s}^{2})$-optimal supersaturated designs (SSDs) with 16 rows and smax=4${s}_{{\rm{\max }}}=4$ is 60 by showing that there exists no resolvable 2-(16, 8, 35) design such that any two blocks from different parallel classes intersect in 3, 5, or 4 points. This is accomplished by an exhaustive computer search that uses the parallel class intersection
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Euclidean designs obtained from spherical embedding of coherent configurations J. Comb. Des. (IF 0.7) Pub Date : 2022-12-19 Aiguo Wang, Yan Zhu
Coherent configurations are a generalization of association schemes. Motivated by the recent study of Q-polynomial coherent configurations, in this paper, we study the spherical embedding of a Q-polynomial coherent configuration into some eigenspace by a primitive idempotent. We present a necessary and sufficient condition when the embedding becomes a Euclidean t$t$-design (on two concentric spheres)
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On the equivalence of certain quasi-Hermitian varieties J. Comb. Des. (IF 0.7) Pub Date : 2022-12-07 Angela Aguglia, Luca Giuzzi
By Aguglia et al., new quasi-Hermitian varieties ◂◽.▸ℳα,β${{\rm{ {\mathcal M} }}}_{\alpha ,\beta }$ in ◂...▸PG(r,q2)$\text{PG}(r,{q}^{2})$ depending on a pair of parameters α,β$\alpha ,\beta $ from the underlying field ◂...▸GF(q2)$\text{GF}({q}^{2})$ have been constructed. In the present paper we study the structure of the lines contained in ◂◽.▸ℳα,β${{\rm{ {\mathcal M} }}}_{\alpha ,\beta }$ and consequently
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Projective planes of order 12 do not have a collineation group of order 4 J. Comb. Des. (IF 0.7) Pub Date : 2022-12-05 Kenzi Akiyama, Chihiro Suetake, Masaki Tanaka
In this paper, we prove that there are no projective planes of order 12 admitting a collineation group of order 4. This yields that the order of any collineation group of a projective plane of order 12 is 1, 2, or 3.
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Tight globally simple nonzero sum Heffter arrays and biembeddings J. Comb. Des. (IF 0.7) Pub Date : 2022-11-15 Lorenzo Mella, Anita Pasotti
Square relative nonzero sum Heffter arrays, denoted by ◂⋅▸NHt(n;k)${\rm{N}}{{\rm{H}}}_{t}(n;k)$, have been introduced as a variant of the classical concept of Heffter array. An ◂⋅▸NHt(n;k)${\rm{N}}{{\rm{H}}}_{t}(n;k)$ is an n×n$n\times n$ partially filled array with elements in Zv${{\mathbb{Z}}}_{v}$, where ◂=▸v=2nk+t$v=2nk+t$, whose rows and whose columns contain k$k$ filled cells, such that
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The existence of irrational most perfect magic squares J. Comb. Des. (IF 0.7) Pub Date : 2022-11-09 Jingyuan Chen, Jinwei Wu, Dianhua Wu
Let n≡◂⋅▸0(mod4)n\unicode{x02261}0(mod4) be a positive integer, ◂=▸M=(mi,j)M=(mi,j) be a magic square, where ◂,▸◂≤▸0≤◂◽.▸mi,j≤n2−1,0≤i,j≤n−10\unicode{x02264}mi,j\unicode{x02264}n2\unicode{x02212}1,0\unicode{x02264}i,j\unicode{x02264}n\unicode{x02212}1. MM is called most perfect magic square (MPMS(n)(n) for short) if ◂=▸◂◽.▸mi,j+m◂,▸i+n2,j+n2=n2−1mi,j\unicode{x0002B}mi\unicode{x0002B}n2,j\unicode
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On large partial ovoids of symplectic and Hermitian polar spaces J. Comb. Des. (IF 0.7) Pub Date : 2022-11-06 Michela Ceria, Jan De Beule, Francesco Pavese, Valentino Smaldore
In this paper we provide constructive lower bounds on the sizes of the largest partial ovoids of the symplectic polar spaces W(3,q)${\mathscr{W}}(3,q)$, q$q$ odd square, q≢◂⋅▸0(mod3)$q\not\equiv 0(\mathrm{mod}3)$, W(5,q)${\mathscr{W}}(5,q)$ and of the Hermitian polar spaces ℋ(4,q2)${\rm{ {\mathcal H} }}(4,{q}^{2})$, q$q$ even or q$q$ odd square, q≢◂⋅▸0(mod3)$q\not\equiv 0(\mathrm{mod}3)$, ℋ(6
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Weak sequenceability in cyclic groups J. Comb. Des. (IF 0.7) Pub Date : 2022-09-26 Simone Costa, Stefano Della Fiore
A subset A$A$ of an abelian group G$G$ is sequenceable if there is an ordering ◂()▸(a1,…,ak)$({a}_{1},\ldots ,{a}_{k})$ of its elements such that the partial sums ◂()▸(s0,s1,…,sk)$({s}_{0},{s}_{1},\ldots ,{s}_{k})$, given by s0=0${s}_{0}=0$ and si=◂∑▸∑j=1iaj${s}_{i}={\sum }_{j=1}^{i}{a}_{j}$ for 1≤i≤k$1\le i\le k$, are distinct, with the possible exception that we may have ◂=▸sk=s0=0${s}_{k}={s}_{0}=0$
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An alternative construction of the Hermitian unital 2-(28, 4, 1) design J. Comb. Des. (IF 0.7) Pub Date : 2022-09-26 Koichi Inoue
In this paper, we give an alternative construction of the Hermitian unital 2-(28, 4, 1) design in such a way that it is constructed on the isotropic vectors in a unitary geometry of dimension 3 over the field F4${{\mathbb{F}}}_{4}$. As a corollary, we can construct a unique 3-(10, 4, 1) design (namely, the Witt system W10${{\boldsymbol{W}}}_{{\bf{10}}}$).
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Completing the spectrum of semiframes with block size three J. Comb. Des. (IF 0.7) Pub Date : 2022-09-13 H. Cao, D. Xu, H. Zheng
A k-semiframe of type gu is a k-GDD of type gu(X,G,ℬ), in which the collection of blocks ℬ can be written as a disjoint union ℬ=P∪Q, where P is partitioned into parallel classes of X and Q is partitioned into holey parallel classes, each holey parallel class being a partition of X\G for some G∈G. In this paper, we will introduce a new concept of t-perfect semiframe and use it to prove the existence
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Stability of Erdős–Ko–Rado theorems in circle geometries J. Comb. Des. (IF 0.7) Pub Date : 2022-08-13 Sam Adriaensen
Circle geometries are incidence structures that capture the geometry of circles on spheres, cones and hyperboloids in three-dimensional space. In a previous paper, the author characterised the largest intersecting families in finite ovoidal circle geometries, except for Möbius planes of odd order. In this paper we show that also in these Möbius planes, if the order is greater than 3, the largest intersecting
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On flag-transitive 2-(k2,k,λ) $({k}^{2},k,\lambda )$ designs with λ∣k $\lambda | k$ J. Comb. Des. (IF 0.7) Pub Date : 2022-07-29 Alessandro Montinaro, Eliana Francot
It is shown that, apart from the smallest Ree group, a flag-transitive automorphism group G$G$ of a 2-(k2,k,λ)$({k}^{2},k,\lambda )$ design D${\mathscr{D}}$, with λ∣k$\lambda | k$, is either an affine group or an almost simple classical group. Moreover, when G$G$ is the smallest Ree group, D${\mathscr{D}}$ is isomorphic either to the 2-(62,6,2)$({6}^{2},6,2)$ design or to one of the three 2-(62,6,6)$({6}^{2}
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The spectrum for large sets of resolvable idempotent Latin squares J. Comb. Des. (IF 0.7) Pub Date : 2022-07-27 Xiangqian Li, Yanxun Chang
An idempotent Latin square of order v$v$ is called resolvable and denoted by RILS(v) if the v(v−1)$v(v-1)$ off-diagonal cells can be resolved into v−1$v-1$ disjoint transversals. A large set of resolvable idempotent Latin squares of order v$v$, briefly LRILS(v), is a collection of v−2$v-2$ RILS(v)s pairwise agreeing on only the main diagonal. In this article, an LRILS(v) is constructed for v∈{14,20
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Extended near Skolem sequences, Part III J. Comb. Des. (IF 0.7) Pub Date : 2022-07-25 Catharine A. Baker, Vaclav Linek, Nabil Shalaby
A k$k$-extended q$q$-near Skolem sequence of order n$n$, denoted by Nnq(k)${{\mathscr{N}}}_{n}^{q}(k)$, is a sequence s1,s2,…,s2n−1${s}_{1},{s}_{2},\ldots ,{s}_{2n-1}$ where sk=0${s}_{k}=0$ and for each integer ℓ∈[1,n]\{q}$\ell \in [1,n]\backslash \{q\}$ there are two indices i$i$, j$j$ such that si=sj=ℓ${s}_{i}={s}_{j}=\ell $ and ∣i−j∣=ℓ$| i-j| =\ell $. For an Nnq(k)${{\mathscr{N}}}_{n}^{q}(k)$ to
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Embedding in MDS codes and Latin cubes J. Comb. Des. (IF 0.7) Pub Date : 2022-06-20 Vladimir N. Potapov
An embedding of a code is a mapping that preserves distances between codewords. We prove that any code with code distance dd$d$ and length nn$n$ can be embedded into an maximum distance separable (MDS) code with the same code distance and length but under a larger alphabet. As a corollary we obtain embeddings of systems of partial mutually orthogonal Latin cubes and nn$n$ -ary quasigroups.
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New infinite classes of 2-chromatic Steiner quadruple systems J. Comb. Des. (IF 0.7) Pub Date : 2022-05-19 Lijun Ji, Shuangqing Liu, Ye Yang
In 1971, Doyen and Vandensavel gave a special doubling construction that gives a direct construction of 2-chromatic Steiner quadruple system of order vv$v$ (SQS(v)(v)$(v)$ ) for all v≡4v≡4$v\equiv 4$ or 8(mod12)8(mod12)$8\,(\mathrm{mod}\,12)$ . The first author presented a construction for 2-chromatic SQSs based on 2-chromatic candelabra quadruple systems and ss$s$ -fan designs. In this paper, it is
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An update on the existence of Kirkman triple systems with steiner triple systems as subdesigns J. Comb. Des. (IF 0.7) Pub Date : 2022-05-16 Peter J. Dukes, Esther R. Lamken
A Kirkman triple system of order vv , KTS(v)(v) , is a resolvable Steiner triple system on vv elements. In this paper, we investigate an open problem posed by Doug Stinson, namely the existence of KTS(v)(v) which contain as a subdesign a Steiner triple system of order uu , an STS(u)(u) . We present several different constructions for designs of this form. As a consequence, we completely settle the
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The existence of partitioned balanced tournament designs J. Comb. Des. (IF 0.7) Pub Date : 2022-05-16 Makoto Araya, Naoya Tokihisa
E. R. Lamken proved that there exists a partitioned balanced tournament design of side nn$n$ , PBTD(nn$n$ ), for nn$n$ a positive integer, n≥5n≥5$n\ge 5$ , except possibly for n∈{9,11,15}n∈{9,11,15}$n\in \{9,11,15\}$ . In this article, we establish the existence of PBTD(nn$n$ ) for n∈{9,11,15}n∈{9,11,15}$n\in \{9,11,15\}$ . As a consequence, the existence of PBTD(nn$n$ ) has now been completely determined
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Algorithms and complexity for counting configurations in Steiner triple systems J. Comb. Des. (IF 0.7) Pub Date : 2022-04-18 Daniel Heinlein, Patric R. J. Östergård
Steiner triple systems form one of the most studied classes of combinatorial designs. Configurations, including subsystems, play a central role in the investigation of Steiner triple systems. With sporadic instances of small systems, ad hoc algorithms for counting or listing configurations are typically fast enough for practical needs, but with many systems or large systems, the relevance of computational
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Classification of minimal blocking sets in small Desarguesian projective planes J. Comb. Des. (IF 0.7) Pub Date : 2022-03-31 Kris Coolsaet, Arne Botteldoorn, Veerle Fack
A full classification (up to equivalence) of all minimal blocking sets in Desarguesian projective planes of order ≤8≤8 was obtained by computer. The resulting numbers of minimal blocking sets are tabulated according to size of the set and order of the automorphism group. For the minimal blocking sets with the larger automorphism groups explicit descriptions are given. Some of these results can also
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On a relation between bipartite biregular cages, block designs and generalized polygons J. Comb. Des. (IF 0.7) Pub Date : 2022-03-23 Gabriela Araujo-Pardo, Robert Jajcay, Alejandra Ramos-Rivera, Tamás Szőnyi
A bipartite biregular (𝑚,𝑛;𝑔)(m,n;g)$(m,n;g)$ -graph ΓΓ${\rm{\Gamma }}$ is a bipartite graph of even girth 𝑔g$g$ having the degree set {𝑚,𝑛}{m,n}$\{m,n\}$ and satisfying the additional property that the vertices in the same partite set have the same degree. An (𝑚,𝑛;𝑔)(m,n;g)$(m,n;g)$ -bipartite biregular cage is a bipartite biregular (𝑚,𝑛;𝑔)(m,n;g)$(m,n;g)$ -graph of minimum order. In their