On q-ary Bent and Plateaued Functions
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We obtain the following results. For any prime q the minimal Hamming distance between distinct regular q-ary bent functions of 2n variables is equal to q^n. The number of q-ary regular bent functions at the distance q^n from the quadratic bent function Q_n=x_1x_2+...+x_2n-1x_2n is equal to q^n(q^n-1+1)...(q+1)(q-1) for q>2. The Hamming distance between distinct binary s-plateaued functions of n variables is not less than 2^s+n-2/2 and the Hamming distance between distinctternary s-plateaued functions of n variables is not less than 3^s+n-1/2. These bounds are tight. For q=3 we prove an upper bound on nonlinearity of ternary functions in terms of their correlation immunity. Moreover, functions reaching this bound are plateaued. For q=2 analogous result are well known but for large q it seems impossible. Constructions and some properties of q-ary plateaued functions are discussed.
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