Properties of Codes from Difference Sets in 2-Groups

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A (v, k, λ)-difference set D in a group G can be used to create a symmetric 2-(v, k, λ) 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, CC = JC, where J is the radical of the group algebra of G over Z2. When G is a 2-group, it is shown that CC is equivalent to the first-order Reed-Muller code, ℛ(1, 2s + 2), precisely when the 2-divisor of CC 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 CC 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 CC is maximal.

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