Bent functions at the minimal distance and algorithms of constructing linear codes for CDMA

In this paper we study linear codes for CDMA (Code Division Multiple Access). This is the standard for the 3rd Generation cellular communications systems. In this standard bent functions are used for constructing codes of constant amplitude. This application allows to decrease PAPR (peak-to-average power ratio) coefficient as much as possible. Such codes consist of vectors of values for bent functions. Let us give the paper structure. In this section we consider some definitions and facts about bent functions. In the second section we give some information about affine equivalent bent functions. In the third section we give a simple method for constructing bent functions at the minimal distance from the given one. In the fourth section we briefly discuss CDMA. In the fifth section we give some known facts on linear codes for CDMA and present a new algorithm for constructing such codes. Our codes of small lengths (obtained with the algorithm) have the best known parameters. Let us give some definitions and known facts connected to bent functions. Denote by dist(f, g) = |{x : f(x) 6= g(x), x ∈ En}| Hamming distance between Boolean functions f and g. Denote by En a n-ary binary cube. Let Fn be a set of all Boolean functions in n variables. By ⊕ denote the sum modulo 2.