Super-simple Steiner pentagon systems

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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“…These HTSPSs can be obtained by Lemma 4.1, inflating super-simple HTSPSs of types 4 6 , 5 6 , and 5 8 (from Lemmas 3.3 and 3.4) with a TD (5,4).…”

Section: Direct Constructions For Tspss and Htspssmentioning

confidence: 99%

“…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%

Super-Simple Twofold Steiner Pentagon Systems

Abel

Bennett

2011

Graphs and Combinatorics

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“…If there exists a (v, 4, q)-CRSS design with b blocks, then there exists a (v, 6, 4) q+1 code of size b; if there exists a (4, q)-CRSSGDD of type g t 1 1 . .…”

Section: Lemma 24 ([66])mentioning

confidence: 99%

“…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.

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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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“…Chen and Wei constructed super-simple (v, 5, 2)-BIBDs for v = 75, 95, 385 in [15]. In [1], Abel and Bennett established existence results for super-simple SPS(v) with just one possible exception, v = 25, and obtained solutions for all the unknown super-simple (v, 5, 2) BIBDs in Theorem 1.1. For convenience, we incorporate these in the following theorem:…”

Section: Introductionmentioning

confidence: 99%

Super‐simple holey Steiner pentagon systems and related designs

Abel

Bennett

2007

J of Combinatorial Designs

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A Steiner pentagon system of order v (SPS(v)) is said to be super-simple if its underlying (v, 5, 2)-BIBD is super-simple; that is, any two blocks of the BIBD intersect in at most two points. It is well known that the existence of a holey Steiner pentagon system (HSPS) of type T implies the existence of a (5, 2)-GDD of type T. We shall call an HSPS of type T super-simple if its underlying (5, 2)-GDD of type T is super-simple; that is, any two blocks of the GDD intersect in at most two points. The existence of HSPSs of uniform type h n has previously been investigated by the authors and others. In this article, we focus our attention on the existence of super-simple HSPSs of uniform type h n .

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Super-simple Steiner pentagon systems

Cited by 30 publications

References 11 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

Super‐simple holey Steiner pentagon systems and related designs

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