On (p a , p, p a , p a?1 )-relative difference sets

L., S.; Schmidt, Bernhard

doi:10.1007/bf01390771

Cited by 26 publications

(16 citation statements)

References 20 publications

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“…Indeed, Pott [48] calls the central problem for these RDSs``one of the most interesting questions about RDSs.'' Ma and Schmidt [37] have recently solved the central problem for r=1 in the abelian case, with some exceptions when w and p are odd, but describe the case r>1 as``much more difficult. '' In this paper we present a recursive construction for semi-regular RDSs in a group G relative to a subgroup U$Z r p , where the Sylow p-subgroup of G has rank at least 2r.…”

Section: D+2mentioning

confidence: 99%

“…The only difference is that here G d, c ÂU d, c has order p (d+c&1) r+(r+i) rather than p (d+c&1) r+i . Since the replacement of conditions on j in the proof of Corollary 7.7 does not rely on the inequality i<r, this part of the proof carries over completely with r+i used instead of each occurrence of i. K [37]. Davis [11] showed that the exponent bound of p d+1 in Corollary 8.2(i) is necessary, and Ma and Pott [36] established the corresponding bound for Corollary 8.2(ii).…”

mentioning

confidence: 96%

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A Unifying Construction for Difference Sets

Davis

Jedwab

1997

Journal of Combinatorial Theory, Series A

105

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We present a recursive construction for difference sets which unifies the Hadamard, McFarland, and Spence parameter families and deals with all abelian groups known to contain such difference sets. The construction yields a new family of difference sets with parameters (v, k, *, n)=(2 2d+4 (2 2d+2 &1)Â3, 2 2d+1 (2 2d+3 +1)Â3, 2 2d+1 (2 2d+1 +1)Â3, 2 4d+2 ) for d 0. The construction establishes that a McFarland difference set exists in an abelian group of order 2 2d+3 (2 2d+1 +1)Â3 if and only if the Sylow 2-subgroup has exponent at most 4. The results depend on a second recursive construction, for semi-regular relative difference sets with an elementary abelian forbidden subgroup of order p r . This second construction deals with all abelian groups known to contain such relative difference sets and significantly improves on previous results, particularly for r>1. We show that the group order need not be a prime power when the forbidden subgroup has order 2. We also show that the group order can grow without bound while its Sylow p-subgroup has fixed rank and that this rank can be as small as 2r. Both of the recursive constructions generalise to nonabelian groups. Academic Press

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Section: D+2mentioning

confidence: 99%

mentioning

confidence: 96%

A Unifying Construction for Difference Sets

Davis

Jedwab

1997

Journal of Combinatorial Theory, Series A

105

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“…As we will show below, it can be also constructed from the Coulter-Matthews bent functions, hence it is now, using Theorem 1, a member of an infinite family. For the two examples, the 3-rank of the adjacency matrices, and the orders of the automorphism groups of the SRGs are (19, 116 640) (affine polar graphs) and (19,5832) (Helleseth-Hamada).…”

Section: Results 7 (Seementioning

confidence: 99%

“…There are many constructions of the RDSs, see [19] for instance. Using MAGMA, we found no RDS constructed from [19, Theorems 2.1, 2.2] which is weakly regular.…”

mentioning

confidence: 99%

Strongly regular graphs associated with ternary bent functions

Tan

Pott

Feng

2010

Journal of Combinatorial Theory, Series A

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We prove a new characterization of weakly regular ternary bent functions via partial difference sets. Partial difference sets are combinatorial objects corresponding to strongly regular graphs. Using known families of bent functions, we obtain in this way new families of strongly regular graphs, some of which were previously unknown. One of the families includes an example in [N. Hamada, T. Helleseth, A characterization of some {3v 2 + v 3 , 3v 1 + v 2 , 3, 3}-minihypers and some [15, 4, 9; 3]-codes with B 2 = 0, J. Statist.Plann. Inference 56 (1996) 129-146], which was considered to be sporadic; using our results, this strongly regular graph is now a member of an infinite family. Moreover, this paper contains a new proof that the Coulter-Matthews and ternary quadratic bent functions are weakly regular.

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“…In their paper [3], Davis and Jedwab improved Ma-Schmidt's result on (p a , p, p a , p a−1 )-difference sets [5] and showed the following. Result 1.1.…”

Section: Introductionmentioning

confidence: 88%

Proof of a conjecture on semiregular relative difference sets in Zp2×Zp2

Hiramine

2006

Journal of Combinatorial Theory, Series A

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On (p a , p, p a , p a?1 )-relative difference sets

Cited by 26 publications

References 20 publications

A Unifying Construction for Difference Sets

A Unifying Construction for Difference Sets

Strongly regular graphs associated with ternary bent functions

Proof of a conjecture on semiregular relative difference sets in Zp2×Zp2

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