Existence of orthogonal latin squares with aligned subsquares

Heinrich, Katherine; Zhu, L.

doi:10.1016/0012-365x(86)90070-1

Cited by 48 publications

(41 citation statements)

References 9 publications

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“…The lower bound comes from Theorem and Proposition , the upper bound is trivial. Proposition () For every integers

q, ℓ \notin {1, 2, 6}

such that

ℓ \leq q / 3

, there exists a pair of orthogonal latin squares of order q with orthogonal latin subsquares of order ℓ.…”

Section: A Lower Bound For the Number Of Mds Codesmentioning

confidence: 99%

On the number of SQSs, latin hypercubes and MDS codes

Potapov

2018

J of Combinatorial Designs

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We establish that the logarithm of the number of latin d‐cubes of order n is Θ(ndlnn) and the logarithm of the number of sets of t (t≥2 is fixed) orthogonal latin squares of order n is Θ(n2lnn). Similar estimations are obtained for systems of mutually strongly orthogonal latin d‐cubes. As a consequence, we construct a set of Steiner quadruple systems of order n such that the logarithm of its cardinality is Θ(n3lnn) as n→∞ and n mod 6=2 or 4.

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“…The lower bound comes from Theorem and Proposition , the upper bound is trivial. Proposition () For every integers

q, ℓ \notin {1, 2, 6}

such that

ℓ \leq q / 3

, there exists a pair of orthogonal latin squares of order q with orthogonal latin subsquares of order ℓ.…”

Section: A Lower Bound For the Number Of Mds Codesmentioning

confidence: 99%

On the number of SQSs, latin hypercubes and MDS codes

Potapov

2018

J of Combinatorial Designs

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“…The 13 blocks of a 2- (15,4,1) packing with a leave containing 2 • K 4 are {1, 8,12,13},{6,8,11,14},{4,6,9,15},{3,7,8,9},{2,8,10,15},{2,9,13,14},{4,5,7,14},{1,6,7,10},{1,5,11,15},{2,7,11,12},{4,10,11,13},{3,12,14,15},{5,9,…”

Section: The Case N ≡mentioning

confidence: 99%

“…Proof. Without loss of generality, we may take the vertex set and edge set of the CP(4) as [8] and {A ⊂ [8] : |A| = 2} \ {{i, i + 4} : i ∈ [4]}, respectively. Consider the subsets of edges E 1 = {A ⊂ [4] : |A| = 2} ∪ ({A ⊂ [3,6] : |A| = 2} \ {{3, 4}}) and E 2 = {A ⊂ {1, 6, 7, 8} : |A| = 2}.…”

Section: Introductionmentioning

confidence: 99%

On Extremal k-Graphs Without Repeated Copies of 2-Intersecting Edges

Chee¹,

Ling²

2007

SIAM J. Discrete Math.

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Abstract. The problem of determining extremal hypergraphs containing at most r isomorphic copies of some element of a given hypergraph family was first studied by Boros et al. in 2001. There are not many hypergraph families for which exact results are known concerning the size of the corresponding extremal hypergraphs, except for those equivalent to the classical Turán numbers. In this paper, we determine the size of extremal k-uniform hypergraphs containing at most one pair of 2-intersecting edges for k ∈ {3, 4}. We give a complete solution when k = 3 and an almost complete solution (with eleven exceptions) when k = 4.

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“…It is known [4] that a pair of orthogonal latin squares of order n can be embedded in a pair of orthogonal latin squares of order t if t ! 3n, the bound of 3n being best possible.…”

Section: Introductionmentioning

confidence: 99%