Properties of Codes from Difference Sets in 2-Groups

A ( v, k, λ)-difference set D in a group G can be used to create a symmetric 2-( v, k, λ) design, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{D}$$ \end{document}, from which arises a code C, generated by vectors corresponding to the characteristic function of blocks of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{D}$$ \end{document}. This paper examines properties of the code C, and of a subcode, Co=JC, where J is the radical of the group algebra of G over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{Z}_2 $$ \end{document}. When G is a 2-group, it is shown that Co is equivalent to the first-order Reed-Muller code, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{R}(1,2s + 2)$$ \end{document}, precisely when the 2-divisor of Co is maximal. In addition, ifD is a non-trivial difference set in an elementary abelian 2-group, and if D is generated by a quadratic bent function, then Co is equal to a power of the radical. Finally, an example is given of a difference set whose characteristic function is not quadratic, although the 2-divisor of Co is maximal.