Triplewhist tournaments that are also Mendelsohn designs

Anderson, Ian; Finizio, Norman J.

doi:10.1002/(sici)1520-6610(1997)5:6<397::aid-jcd1>3.0.co;2-a

Cited by 15 publications

(45 citation statements)

References 5 publications

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“…A modern account of Kirkman's work is given in [1]. From the preceding discussion, it is easy to deduce that a 2-chromatic S(2, 4, v) having all blocks containing three points of one colour and one of the other colour can exist only if v is of the form (12s + 2) 2 or (12s + 10) 2 , s ≥ 0.…”

Section: A Steiner System S(t K V) Is An Ordered Pair (V B)mentioning

confidence: 99%

The design of the century

Forbes

Grannell

Griggs

2007

Mathematica Slovaca

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ABSTRACT. We construct a 2-chromatic Steiner system S(2, 4, 100) in which every block contains three points of one colour and one point of the other colour.The existence of such a design has been open for over 25 years. Steiner system S(t, k, v) is an ordered pair (V, B) where V is a set of cardinality v, the base set, and B is a collection of k-subsets of V , the blocks, which collectively have the property that every t-element subset of V is contained in precisely one block. Elements of V are called points. In this paper we are principally concerned with the case in which t = 2 and k = 4. Steiner systems S(2, 4, v) exist if and only if v ≡ 1 or 4 (mod 12), [4]; such values of v are called admissible. Given a Steiner system S(2, 4, v), we may ask whether it is possible to colour each point of the base set V with one of two colours, say red or blue, so that no block is monochromatic. A Steiner system S(2, 4, v) having this property is said to be 2-chromatic or to have a blocking set. It was shown in [5] that 2-chromatic S(2, 4, v)s exist for all admissible v with the possible exception of three values, v = 37, 40 and 73. Existence for these three values was established in [3]. Perhaps we should also remark here that it is known that for all v ≥ 25 there exists a Steiner system S(2, 4, v) which is not 2-chromatic, [8].In a 2-chromatic S(2, 4, v) let c and v − c be the cardinalities of the red and blue colour classes, respectively. If b 1 , b 2 and b 3 are the numbers of blocks with 2000 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 05B05. K e y w o r d s: Steiner system, colouring, 2-chromatic, blocking set.

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Section: A Steiner System S(t K V) Is An Ordered Pair (V B)mentioning

confidence: 99%

The design of the century

Forbes

Grannell

Griggs

2007

Mathematica Slovaca

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“…The LDPC codes satisfying these two requirements in [4] were designed by exhaustive computer search, in [5] they were designed as codes over GF(4) by identifying the Pauli operators I, X, Y, Z with elements from GF(4), while in [6] they were designed in quasi-cyclic fashion. In what follows, we will show how to design the dual-containing LDPC codes using the combinatorial objects known as BIBDs [7]. Notice that the theory behind BIBDs is well known (see [7]), and BIBDs of unity index have already been used to design LDPC codes of girth-6 [8].…”

Section: Quantum Ldpc Codes From Bibdsmentioning

confidence: 99%

“…In what follows, we will show how to design the dual-containing LDPC codes using the combinatorial objects known as BIBDs [7]. Notice that the theory behind BIBDs is well known (see [7]), and BIBDs of unity index have already been used to design LDPC codes of girth-6 [8]. Notice, however, that dual-containing LDPC are girth-4 LDPC codes, and they can be designed based on BIBDs with even index.…”

Section: Quantum Ldpc Codes From Bibdsmentioning

confidence: 99%

“…In this paper we propose to design quantum LDPC codes using the combinatorial concepts [7]. It was recently shown in [8] that combinatorial constructions can successfully be used to design good classical LDPC codes.…”

Section: Introductionmentioning

confidence: 99%

“…We propose a series of quantum LDPC codes designed using the combinatorial object known as balanced incomplete block design (BIBD) [7]. The quantum LDPC codes, introduced in this letter, have the high code rate (around 0.9).…”

Section: Introductionmentioning

confidence: 99%

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Quantum LDPC Codes from Balanced Incomplete Block Designs

Djordjević

2008

IEEE Commun. Lett.

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Abstract-We present a series of structured LDPC codes suitable for use in quantum error correction. Those codes belong to the class of dual-containing Calderbank-Shor-Steane (CSS) codes. The component CSS code is designed using the combinatorial object known as balanced incomplete block design (BIBD) with an even index. The quantum LDPC codes have the rate around 0.9. Several examples of quantum LDPC codes from BIBDs from unity index, extended by addition of an allones column, are introduced as well. To improve the BER performance, we employed the method of removing the cycles of length four in corresponding bipartite graph.

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Ordered tournaments and ordered triplewhist tournaments with the three person property

Abel

2008

J of Combinatorial Designs

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Triplewhist tournaments that are also Mendelsohn designs

Cited by 15 publications

References 5 publications

The design of the century

The design of the century

Quantum LDPC Codes from Balanced Incomplete Block Designs

Ordered tournaments and ordered triplewhist tournaments with the three person property

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