Super-simple balanced incomplete block designs with block size 4 and index 5

Cao, Haitao; Chen, Kejun; Wei, Ruizhong

doi:10.1016/j.disc.2008.07.003

Cited by 27 publications

(39 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These conditions are also sufficient, except for (h, n) = (1,5), (1,15), (2,5) and possibly for (h, n) ∈ {(1, 25), (13,15), (17,15), (19,15)}.…”

Section: Theorem 23 Necessary Conditions For a Super-simple Hsps Of mentioning

confidence: 92%

“…Here, we found that our designs are more likely to have the super-simple property if several blocks contain at least one repeated difference (this usually occurs when three points in the block are of the form 0, x, and 2x for some x ∈ G). (0, 7, 5, 22, 13), (0, 18,13,9,20), (0, 11,19,20,16), (13,6,3,24,7), (0, 3,7,25,19), (13,20,6,16,12), (0, 8,14,13,24), (0, 13, 1, 11, 21), (0, 1, 24, 3, 12), (13,16,14,25,11) +2 mod 26 v = 30: ( 0, 37, 46, 42, 74), (0, 59, 80, 57, 76), (0, 56, 40, 69, 70), (0, 19, 9, 4, 34) mod 81 v = 85: (0, 72, 51, 57, 49), (0, 11, 48, 68, 25), (0, 25, 58, 41, 57), (0, 10, 15, 75, 45), 0, 36, 79, 95, 101), (0, 16, 97, 67, 26), (0, 42, 15, 122, 105),(0, 7, 20, 101, 10), (0, 19, 18, 91, 100), (0, 32, 1, 116, 33), (0, 71, 12, 75, 127), (0, 77, 35, 68, 39), (0, 71, 111, 105, 11), (0, 51, 26 In the construction of GDDs or PBDs, the technique of "filling in holes" plays an important role. The technique for super-simple TSPSs is described as follows.…”

Section: Direct Constructions For Tspss and Htspssmentioning

confidence: 99%

“…The case λ = 4 was solved independently by Adams et al [7] and Chen [15]. The cases λ = 5 and 6 were solved by Chen, Cao and Wei [13,16]. A recent survey on super-simple (v, 4, λ)-BIBDs with v ≤ 32 can be found in [12].…”

mentioning

confidence: 97%

“…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). …”

mentioning

confidence: 99%

See 3 more Smart Citations

Super-Simple Twofold Steiner Pentagon Systems

Abel

Bennett

2011

Graphs and Combinatorics

View full text Add to dashboard Cite

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.

show abstract

“…These conditions are also sufficient, except for (h, n) = (1,5), (1,15), (2,5) and possibly for (h, n) ∈ {(1, 25), (13,15), (17,15), (19,15)}.…”

Section: Theorem 23 Necessary Conditions For a Super-simple Hsps Of mentioning

confidence: 92%

Section: Direct Constructions For Tspss and Htspssmentioning

confidence: 99%

mentioning

confidence: 97%

mentioning

confidence: 99%

See 2 more Smart Citations

Super-Simple Twofold Steiner Pentagon Systems

Abel

Bennett

2011

Graphs and Combinatorics

View full text Add to dashboard Cite

show abstract

“…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%

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

Zhang

2010

Des. Codes Cryptogr.

View full text Add to dashboard Cite

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}.

show abstract

Decomposable super‐simple NRBIBDs with block size 4 and index 6

Sun

et al. 2018

J of Combinatorial Designs

View full text Add to dashboard Cite

Necessary conditions for the existence of a super‐simple, decomposable, near‐resolvable ( v , 4 , 6 )‐balanced incomplete block design (BIBD) whose 2‐component subdesigns are both near‐resolvable ( v , 4 , 3 )‐BIBDs are v ≡ 1 (mod 4) and v ≥ 17. In this paper, we show that these necessary conditions are sufficient. Using these designs, we also establish that the necessary conditions for the existence of a super‐simple near‐resolvable ( v , 4 , 3 )‐RBIBD, namely v ≡ 1 (mod 4) and v ≥ 9, are sufficient. A few new pairwise balanced designs are also given.

show abstract

Super-simple balanced incomplete block designs with block size 4 and index 5

Cited by 27 publications

References 12 publications

Super-Simple Twofold Steiner Pentagon Systems

Super-Simple Twofold Steiner Pentagon Systems

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

Decomposable super‐simple NRBIBDs with block size 4 and index 6

Contact Info

Product

Resources

About