Super-simple (v,5,2)-designs

Gronau, Hans-Dietrich O. F.; Kreher, Donald L.; Ling, Alan C. H.

doi:10.1016/s0166-218x(03)00270-1

Cited by 34 publications

(46 citation statements)

References 7 publications

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“…For example, such designs are used in constructing perfect hash families [56] and coverings [10], in the construction of new designs [9] and in the construction of superimposed codes [45]. Much attention has been paid on the existence of super-simple BIBDs with block size four and index λ ∈ {2, 3, 4, 5, 6, 9} [5,12,15,16,21,40,41,44], or block size five and index λ ∈ {2, 4, 5} [1,4,17,18,42]. However, nothing has been done for the existence of super-simple packings, which have been shown to be closely related to optimal CWCs [66].…”

Section: Lemma 11 ([5])mentioning

confidence: 99%

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Completely reducible super-simple designs with block size four and related super-simple packings

Zhang

2010

Des. Codes Cryptogr.

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A design is said to be super-simple if the intersection of any two blocks has at most two elements. A super-simple design D with point set X , block set B and index λ is called completely reducible super-simple (CRSS), if its block set B can be written as B = λ i=1 B i , such that B i forms the block set of a design with index unity but having the same parameters as D for each 1 ≤ i ≤ λ. It is easy to see, the existence of CRSS designs with index λ implies that of CRSS designs with index i for 1 ≤ i ≤ λ. CRSS designs are closely related to q-ary constant weight codes (CWCs). A (v, 4, q)-CRSS design is just an optimal (v, 6, 4) q+1 code. On the other hand, CRSS group divisible designs (CRSSGDDs) can be used to construct q-ary group divisible codes (GDCs), which have been proved useful in the constructions of q-ary CWCs. In this paper, we mainly investigate the existence of CRSS designs. Three neat results are obtained as follows. Firstly, we determine completely the spectrum for a (v, 4, 3)-CRSS design. As a consequence, a class of new optimal (v, 6, 4) 4 codes is obtained. Secondly, we give a general construction for (4, 4)-CRSSGDDs with skew Room frames, and prove that the necessary conditions for the existence of a (4, 2)-CRSSGDD of type g u are also sufficient except definitely for (g, u) ∈ {(2, 4), (3, 4), (6, 4)}. Finally, we consider the related optimal super-simple (v, 4, 2)-packings and show that such designs exist for all v ≥ 4 except definitely for v ∈ {4, 5, 6, 9}.

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Section: Lemma 11 ([5])mentioning

confidence: 99%

“…For v ∈ {33, 42, 45}, take the super-simple (v, w; 4, 2)-packings constructed in Lemma 6.10 for (v, w) ∈ {(33, 7), (42,10), (45,13)}, then fill in the holes with optimal super-simple (w, 4, 2)-packings for w ∈ {7, 10, 13} (see Lemma 6.3), respectively, to obtain the required designs.…”

Section: Proofmentioning

confidence: 99%

Completely reducible super-simple designs with block size four and related super-simple packings

Zhang

2010

Des. Codes Cryptogr.

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“…Recently, Gronau et al [12] solved the case of k = 5 and λ = 2 with 11 unsettled values. They showed the following.…”

Section: Introductionmentioning

confidence: 99%

Super-simple (ν, 5, 5) Designs

Chen

Wei

2006

Des Codes Crypt

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Super-simple designs are useful in constructing codes and designs such as superimposed codes and perfect hash families. In this article, we investigate the existence of a super-simple (v, 5, 5) balanced incomplete block design and show that such a design exists if and only if v ≡ 1 (mod 4) and v ≥ 17 except possibly when v = 21. Applications of the results to optical orthogonal codes are also mentioned.

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“…Develop the given blocks (mod 5n) for n = 6 and +2 (mod 30) for n = 6. n = 6: (0, 21,11,26,25), (15,26,10,6,11), (0, 9,13,16,17), (15,1,28,24,2), (0, 2,27,19,16), (15,1,12,17,4), (0, 7, 10, 2, 17), (15,17,25,2,22), (0, 20,21,19,28), (15,4,13,6,5). …”

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confidence: 99%

Super-Simple Twofold Steiner Pentagon Systems

Abel

Bennett

2011

Graphs and Combinatorics

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A twofold pentagon system of order v is a decomposition of the complete undirected 2-multigraph 2K v into pentagons. A twofold Steiner pentagon system of order v [TSPS(v)] is a twofold pentagon system such that every pair of distinct vertices is joined by a path of length two in exactly two pentagons of the system. A TSPS(v) is said to be super-simple if its underlying (v, 5, 4)-BIBD is super-simple; that is, if any two blocks of the BIBD intersect in at most two points. In this paper, it is shown that the necessary conditions for the existence of a super-simple TSPS(v); namely, v ≥ 15 and v ≡ 0 or 1 (mod 5) are sufficient. For these specified orders, the main result of this paper also guarantees the existence of a very special and interesting class of twofold and fourfold Steiner pentagon systems of order v with the additional property that, for any two vertices, the two or four paths of length two joining them are distinct.

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Super-simple (v,5,2)-designs

Cited by 34 publications

References 7 publications

Completely reducible super-simple designs with block size four and related super-simple packings

Completely reducible super-simple designs with block size four and related super-simple packings

Super-simple (ν, 5, 5) Designs

Super-Simple Twofold Steiner Pentagon Systems

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