Almost difference sets in nonabelian groups

We give two new constructions of almost difference sets. The first is a generic construction of (q2(q+1),q(q2-1),q(q2-q-1),q2-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(q^{2}(q+1),\,q(q^{2}-1),\,q(q^{2}-q-1),\,q^{2}-1)$$\end{document} almost difference sets in certain groups of order q2(q+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q^{2}(q+1)$$\end{document} (q is an odd prime power) having (Fq2,+)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_{q^{2}},\,+)$$\end{document} as a subgroup. This construction yields several infinite families of almost difference sets, many of which occur in nonabelian groups. The second construction yields (4p,2p+1,p,p-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(4p,\,2p+1,\,p,\,p-1)$$\end{document} almost difference sets in dihedral groups of order 4p where p≡3(mod4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\equiv 3 \ (\mathrm{mod} \ 4)$$\end{document} is a prime. Moreover, it turns out that some of the infinite families of almost difference sets obtained produce Cayley graphs which are Ramanujan graphs.