Properties of codes from difference sets in 2-groups

A (nu, k, lambda)-difference set D in a group G can be used to create a symmetric 2-(nu, k, lambda) design, D, from which arises a code C, generated by vectors corresponding to the characteristic function of blocks of D. This paper examines properties of the code C, and of a subcode, C-0 = JC, where J is the radical of the group algebra of G over Z(2). When G is a 2-group, it is shown that C-0 is equivalent to the first-order Reed-Muller code, R(1, 2s + 2), precisely when the 2-divisor of C-0 is maximal. In addition, if D is a non-trivial difference set in an elementary abelian 2-group, and if D is generated by a quadratic bent function, then C-0 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 C-0 is maximal.