Construction of relative difference sets and Hadamard groups

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.