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Construction of relative difference sets and Hadamard groups

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Abstract

‘There exist normal \((2m,2,2m,m)\) relative difference sets and thus Hadamard groups of order \(4m\) for all \(m\) of the form

$$\begin{aligned} m= x2^{a+t+u+w+\delta -\epsilon +1}6^b 9^c 10^d 22^e 26^f \prod _{i=1}^s p_i^{4a_i} \prod _{i=1}^t q_i^2 \prod _{i=1}^u \left( (r_i+1)/2)r_i^{v_i}\right) \prod _{i=1}^w s_i \end{aligned}$$

under the following conditions: \(a,b,c,d,e,f,s,t,u,w\) are nonnegative integers, \(a_1,\ldots ,a_r\) and \(v_1,\ldots ,v_u\) are positive integers, \(p_1,\ldots ,p_s\) are odd primes, \(q_1,\ldots ,q_t\) and \(r_1,\ldots ,r_u\) are prime powers with \(q_i\equiv 1\ (\mathrm{mod}\ 4)\) and \(r_i\equiv 1\ (\mathrm{mod}\ 4)\) for all \(i, s_1,\ldots ,s_w\) are integers with \(1\le s_i \le 33\) or \(s_i\in \{39,43\}\) for all \(i, x\) is a positive integer such that \(2x-1\) or \(4x-1\) is a prime power. Moreover, \(\delta =1\) if \(x>1\) and \(c+s>0, \delta =0\) otherwise, \(\epsilon =1\) if \(x=1, c+s=0\), and \(t+u+w>0, \epsilon =0\) otherwise. We also obtain some necessary conditions for the existence of \((2m,2,2m,m)\) relative difference sets in partial semidirect products of \(\mathbb{Z }_4\) with abelian groups, and provide a table cases for which \(m\le 100\) and the existence of such relative difference sets is open.

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Acknowledgments

We thank the referees for the careful reading of the manuscript.

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Authors and Affiliations

  1. Division of Mathematical Sciences, School of Physical & Mathematical Sciences, Nanyang Technological University, Singapore, 637371, Singapore

    Bernhard Schmidt & Ming Ming Tan

Authors
  1. Bernhard Schmidt
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Corresponding author

Correspondence to Bernhard Schmidt.

Additional information

Communicated by K. T. Arasu.

Appendix

Appendix

The following table presents all abelian groups of order divisible by 4, up to 400, which do not admit Golay transversals or for which the existence of such transversals is open. In the case of nonexistence, a reference to the proof is provided. Groups that admit Golay transversals by Theorems 6.1 and 6.2 are not listed.

Order

Group

Exist?

12

(2)(2)(3)

Lem 7.3

24

(2)(2)(2)(3)

Lem 7.3

28

(2)(2)(7)

Lem 7.3

36

(2)(2)(9)

Lem 7.3

44

(2)(2)(11)

Lem 7.3

48

(2)(2)(2)(2)(3)

Lem 7.3

56

(2)(2)(2)(7)

Lem 7.3

60

(2)(2)(3)(5)

Lem 7.2

68

(2)(2)(17)

?

72

(2)(2)(2)(9)

Lem 7.3

76

(2)(2)(19)

Lem 7.3

84

(2)(2)(3)(7)

Lem 7.2

88

(2)(2)(2)(11)

Lem 7.3

92

(2)(2)(23)

Lem 7.3

96

(2)(2)(2)(2)(2)(3)

Lem 7.3

100

(2)(2)(25)

?

100

(2)(2)(5)(5)

?

108

(2)(2)(27)

Lem 7.3

108

(3)(4)(9)

?

108

(2)(2)(3)(9)

Lem 7.3

108

(3)(3)(3)(4)

?

108

(2)(2)(3)(3)(3)

Lem 7.2

112

(2)(2)(2)(2)(7)

Lem 7.3

116

(2)(2)(29)

?

120

(2)(2)(2)(3)(5)

Lem 7.2

124

(2)(2)(31)

Lem 7.3

132

(2)(2)(3)(11)

Lem 7.2

136

(2)(2)(2)(17)

?

140

(2)(2)(5)(7)

Lem 7.2

144

(2)(2)(2)(2)(9)

Lem 7.3

148

(2)(2)(37)

?

152

(2)(2)(2)(19)

Lem 7.3

156

(2)(2)(3)(13)

Lem 7.2

Order

Group

Exist?

164

(2)(2)(41)

?

168

(2)(2)(2)(3)(7)

Lem 7.2

172

(2)(2)(43)

Lem 7.3

176

(2)(2)(2)(2)(11)

Lem 7.3

180

(2)(2)(5)(9)

?

180

(2)(2)(3)(3)(5)

?

184

(2)(2)(2)(23)

Lem 7.3

188

(4)(47)

?

188

(2)(2)(47)

Lem 7.3

192

(2)(2)(2)(2)(2)(2)(3)

Lem 7.3

196

(2)(2)(49)

Cor 3.7

196

(4)(7)(7)

?

196

(2)(2)(7)(7)

Cor 6.3

204

(2)(2)(3)(17)

Lem 7.2

212

(2)(2)(53)

?

216

(2)(2)(2)(27)

Lem 7.3

216

(2)(2)(2)(3)(9)

Lem 7.3

216

(2)(2)(2)(3)(3)(3)

Lem 7.2

220

(2)(2)(5)(11)

Lem 7.2

224

(2)(2)(2)(2)(2)(7)

Lem 7.3

228

(2)(2)(3)(19)

Lem 7.2

232

(2)(2)(2)(29)

?

236

(4)(59)

?

236

(2)(2)(59)

Lem 7.3

240

(2)(2)(2)(2)(3)(5)

Lem 7.2

244

(2)(2)(61)

?

248

(2)(2)(2)(31)

Lem 7.3

252

(2)(2)(7)(9)

Lem 7.2

252

(3)(3)(4)(7)

?

252

(2)(2)(3)(3)(7)

Lem 7.2

260

(4)(5)(13)

?

260

(2)(2)(5)(13)

?

264

(2)(2)(2)(3)(11)

Lem 7.2

268

(4)(67)

?

268

(2)(2)(67)

Lem 7.3

272

(2)(2)(2)(2)(17)

?

276

(2)(2)(3)(23)

Lem 7.2

280

(2)(2)(2)(5)(7)

Lem 7.2

284

(2)(2)(71)

Lem 7.3

288

(2)(2)(2)(2)(2)(9)

Lem 7.3

292

(4)(73)

?

292

(2)(2)(73)

?

Order

Group

Exist?

296

(2)(2)(2)(37)

?

300

(2)(2)(3)(25)

Lem 7.2

300

(2)(2)(3)(5)(5)

Lem 7.2

304

(2)(2)(2)(2)(19)

Lem 7.3

308

(2)(2)(7)(11)

Lem 7.2

312

(2)(2)(2)(3)(13)

Lem 7.2

316

(2)(2)(79)

Lem 7.3

324

(4)(81)

?

324

(2)(2)(81)

Lem 7.3

324

(3)(4)(27)

?

324

(2)(2)(3)(27)

Lem 7.3

324

(3)(3)(4)(9)

?

324

(2)(2)(3)(3)(9)

Cor 7.5

328

(2)(2)(2)(41)

?

332

(2)(2)(83)

Lem 7.3

336

(2)(2)(2)(2)(3)(7)

Lem 7.2

340

(2)(2)(5)(17)

?

344

(2)(2)(2)(43)

Lem 7.3

348

(2)(2)(3)(29)

Lem 7.2

352

(2)(2)(2)(2)(2)(11)

Lem 7.3

356

(4)(89)

?

356

(2)(2)(89)

?

360

(2)(2)(2)(5)(9)

?

364

(2)(2)(7)(13)

Lem 7.2

368

(2)(2)(2)(2)(23)

Lem 7.3

372

(3)(4)(31)

?

372

(2)(2)(3)(31)

Lem 7.2

376

(8)(47)

?

376

(2)(4)(47)

?

376

(2)(2)(2)(47)

Lem 7.3

380

(2)(2)(5)(19)

Lem 7.2

384

(2)(2)(2)(2)(2)(2)(2)(3)

Lem 7.3

388

(2)(2)(97)

?

392

(2)(2)(2)(49)

Lem 7.3

392

(7)(7)(8)

?

392

(2)(2)(2)(7)(7)

?

396

(2)(2)(9)(11)

Lem 7.2

396

(3)(3)(4)(11)

?

396

(2)(2)(3)(3)(11)

Lem 7.2

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Schmidt, B., Tan, M.M. Construction of relative difference sets and Hadamard groups. Des. Codes Cryptogr. 73, 105–119 (2014). https://doi.org/10.1007/s10623-013-9811-x

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