More large sets of resolvable MTS and resolvable DTS with odd orders

Kang, Qingde; Zhao, Hongtao

doi:10.1016/j.disc.2007.07.032

Cited by 4 publications

(6 citation statements)

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“…By [7], there exists an LRMTS(q + 2) for q ∈ {67, 109, 11 2 , 139, 157, 163, 181, 193, 199}. By [13], an LRMTS(q + 2) exists for q ∈ {211, 229, 277, 283, 17 2 , 307, 337}. It is not difficult to see from the original texts that all these known LRMTS(q + 2)s have quasisymmetric property.…”

Section: Resultsmentioning

confidence: 87%

“…Thus it is desirable to construct an LRQMTS(v) of order v = 381, 399; Ref. [13] serves as a guideline for direct constructions.…”

Section: Resultsmentioning

confidence: 99%

“…After determination of the existence spectrum of LMTSs, some researchers turned attention to such large sets with resolvable property. In a number of articles, [3,7,[10][11][12][13]19,21], the authors presented some construction approaches. In particular, the authors in [7,12,13] investigated direct constructions for LRMTSs, which usually assume an appropriate group of automorphisms for the putative designs and large sets.…”

Section: Introductionmentioning

confidence: 99%

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Large sets of resolvable Mendelsohn triple systems of prime power sizes

Geng

Zhou

2015

Discrete Mathematics

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a b s t r a c tA large set of Mendelsohn triple systems of order v is a partition of all cyclic triples on a v-element set into pairwise disjoint Mendelsohn triple systems of order v. This note addresses questions related to the construction of large sets of Mendelsohn triple systems with resolvable property (each block set having a partition into parallel classes). An improved recursive method is established and a number of new infinite series large sets of prime power sizes are settled by recursion, in combination with already known small cases.To be specific, for all prime powers q < 400 and q ≡ 1 (mod 3), large sets of resolvable Mendelsohn triple systems of order q n + 2 are proved to exist for any positive integer n, possibly except for q = 379, 397.

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Section: Resultsmentioning

confidence: 87%

“…Thus it is desirable to construct an LRQMTS(v) of order v = 381, 399; Ref. [13] serves as a guideline for direct constructions.…”

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Large sets of resolvable Mendelsohn triple systems of prime power sizes

Geng

Zhou

2015

Discrete Mathematics

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“…A large set of disjoint RDTS(v)s is denoted by LRDTS(v). The existence of LRDTS(v)s has been investigated by Kang [8], Kang and Lei [10], Kang and Tian [11], Kang and Xu [12], Kang and Zhao [13], Xu and Kang [17] and Zhou and Chang [22,23]. By their research and related results about large sets of Kirkman triple systems [3,4,5,6,14,15,18,19,20,21], we can list the known results as follows.…”

Section: Introductionmentioning

confidence: 99%

Another Product Construction for Large Sets of Resolvable Directed Triple Systems

Zhao¹

2009

Electron. J. Combin.

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A large set of resolvable directed triple systems of order $v$, denoted by LRDTS$(v)$, is a collection of $3(v-2)$ RDTS$(v)$s based on $v$-set $X$, such that every transitive triple of $X$ occurs as a block in exactly one of the $3(v-2)$ RDTS$(v)$s. In this paper, we use DTRIQ and LR-design to present a new product construction for LRDTS$(v)$s. This provides some new infinite families of LRDTS$(v)$s.

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“…The parallel results to (1) and (2) for large sets of Kirkman triple systems or resolvable Mendelsohn triple systems can be found in [7,13]. Then we apply the product constructions in [15,16] to get more infinite classes, where we utilize the existence of LR-designs in [6].…”

mentioning

confidence: 97%

Large sets of resolvable idempotent Latin squares

Zhou

Chang

Tian

2011

Discrete Mathematics

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a b s t r a c tAn idempotent Latin square of order v is called resolvable and denoted by RILS(v) if the v(v − 1) off-diagonal cells can be resolved into v − 1 disjoint transversals. A large set of resolvable idempotent Latin squares of order v, briefly LRILS(v), is a collection of v − 2 RILS(v)s pairwise agreeing on only the main diagonal. In this paper we display some recursive and direct constructions for LRILSs.A Latin square of order v LS(v) is a v × v array in which each cell contains a single symbol from a v-set X , such that each symbol occurs exactly once in each row and exactly once in each column. We usually index the rows, columns, and symbols(1) each cell of L is empty or contains an element of X ; (2) for each 1 ≤ i ≤ k, the subarray indexed by H i × H i is empty (each H i is called a hole); and (3) the elements in row or column x are exactly those of X \ H i if x ∈ H i , and of X otherwise. If each |H i | = h, 1 ≤ i ≤ k, we denote the incomplete Latin square by LS(v; h k ). An (incomplete) Latin square L = (a ij ) is idempotent if a ii = i for each i not in the holes. An ILS(v) denotes an idempotent LS(v).Let L be an (incomplete) Latin square of order v on a symbol set X . A transversal of L is a set of v cells, one from each row and column, containing each of the v symbols exactly once. If L is an ILS(v), then the main diagonal cells form a transversal, which we call an idempotent transversal.Suppose that L = (a ij ) and L ′ = (b ij ) are LS(v)s on a set X . L and L ′ are orthogonal if every element of X × X occurs exactly once among the v 2 pairs (a ij , b ij ), i, j ∈ X . If an idempotent Latin square L = (a ij ) on I v is resolvable with v disjoint transversals T 0 , T 1 , . . . , T v−1 , then it has an orthogonal mate L ′ = (b ij ), where b ij = k if (i, j) ∈ T k . If T 0 is the idempotent transversal of L, then the main diagonal entries of L ′ are all 0's. For an RILS we often use such an orthogonal mate to designate its resolution.

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More large sets of resolvable MTS and resolvable DTS with odd orders

Cited by 4 publications

References 18 publications

Large sets of resolvable Mendelsohn triple systems of prime power sizes

Large sets of resolvable Mendelsohn triple systems of prime power sizes

Another Product Construction for Large Sets of Resolvable Directed Triple Systems

Large sets of resolvable idempotent Latin squares

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