Construction Techniques for Anti-Pasch Steiner Triple Systems

Ling, Alan C. H.; Colbourn, Charles J.; Grannell, M. J.; Griggs, T. S.

doi:10.1112/s0024610700008838

Cited by 45 publications

(61 citation statements)

References 5 publications

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“…In particular, Brouwer [1] refined the Erdó´s' conjecture for the case r ¼ 4 to assert that an anti-Pasch STSðvÞ exists whenever v 1; 3 (mod 6) except for v ¼ 7 and 13. While this anti-Pasch conjecture had been long standing, combining together with many earlier results, Ling, Colbourn, Grannell and Griggs [9] substantially narrowed the spectrum of possible exceptions and the conjecture was established by Grannell, Griggs and Whitehead [6]. Theorem 1.1 (Grannell, Griggs and Whitehead) [6].…”

Section: Introductionmentioning

confidence: 89%

Infinite classes of anti‐mitre and 5‐sparse Steiner triple systems

Fujiwara

2005

J of Combinatorial Designs

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In this paper, we present three constructions for anti-mitre Steiner triple systems and a construction for 5-sparse ones. The first construction for anti-mitre STSs settles two of the four unsettled admissible residue classes modulo 18 and the second construction covers such a class modulo 36. The third construction generates many infinite classes of anti-mitre STSs in the remaining possible orders. As a consequence of these three constructions we can construct anti-mitre systems for at least 13=14 of the admissible orders. For 5-sparse STSðvÞ, we give a construction for v 1; 19 (mod 54) and v 6 ¼ 109. # 2005 Wiley Periodicals, Inc. J Combin

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Section: Introductionmentioning

confidence: 89%

Infinite classes of anti‐mitre and 5‐sparse Steiner triple systems

Fujiwara

2005

J of Combinatorial Designs

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“…In [19], Colbourn lists 80 nonisomorphicdesigns, only one of which is Pasch-free (see Table II) (this particular design is also described in [19]). In their recent work, Ling et al [46] present a construction techniques for anti-Pasch STSs. However, these designs do not necessarily lead to codes with quasi-cyclic structure.…”

Section: Remark 33mentioning

confidence: 99%

Combinatorial Constructions of Low-Density Parity-Check Codes for Iterative Decoding

Vasić

Milenković

2004

IEEE Trans. Inform. Theory

217

172

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Abstract-This paper introduces several new combinatorial constructions of low-density parity-check (LDPC) codes, in contrast to the prevalent practice of using long, random-like codes. The proposed codes are well structured, and unlike random codes can lend themselves to a very low-complexity implementation. Constructions of regular Gallager codes based on cyclic difference families, cycle-invariant difference sets, and affine 1-configurations are introduced. Several constructions of difference families used for code design are presented, as well as bounds on the minimal distance of the codes based on the concept of a generalized Pasch configuration.

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“…Many partial results had been developed for this conjecture (see Colbourn and Rosa [8]). In particular, by developing several new constructions, Ling, Colbourn, Grannell and Griggs [28] substantially extended the spectrum of 4-sparse Steiner triple systems; and, finally, Brouwer's conjecture was settled by Grannell, Griggs and Whitehead [19]: Theorem 1.3 (Grannell, Griggs and Whitehead [19]) There exists a 4-sparse STS(v) if and only if v ≡ 1, 3 (mod 6) and v = 7, 13.…”

Section: ) Let L(k L) Be the Family Of All Nonisomorphic 3-uniform Hmentioning

confidence: 99%

Nonexistence of sparse triple systems over abelian groups and involutions

Fujiwara

2007

J Algebr Comb

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In 1973 Paul Erdős conjectured that there is an integer v 0 (r) such that, for every v > v 0 (r) and v ≡ 1, 3 (mod 6), there exists a Steiner triple system of order v, containing no i blocks on i + 2 points for every 1 < i ≤ r. Such an STS is said to be r-sparse. In this paper we consider relations of automorphisms of an STS to its sparseness. We show that for every r ≥ 13 there exists no point-transitive r-sparse STS over an abelian group. This bound and the classification of transitive groups give further nonexistence results on block-transitive, flag-transitive, 2-transitive, and 2-homogeneous STSs with high sparseness. We also give stronger bounds on the sparseness of STSs having some particular automorphisms with small groups. As a corollary of these results, it is shown that various well-known automorphisms, such as cyclic, 1-rotational over arbitrary groups, and involutions, prevent an STS from being high-sparse.

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Construction Techniques for Anti-Pasch Steiner Triple Systems

Cited by 45 publications

References 5 publications

Infinite classes of anti‐mitre and 5‐sparse Steiner triple systems

Infinite classes of anti‐mitre and 5‐sparse Steiner triple systems

Combinatorial Constructions of Low-Density Parity-Check Codes for Iterative Decoding

Nonexistence of sparse triple systems over abelian groups and involutions

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