Correlation-immune functions over finite fields

We give a series of constructions of correlation-immune function over finite fields. We prove that F₂ and F₃ are the only finite fields F_q with the property that every (n-1)th correlation-immune function in n>2 variables over F_q is linear. We also show that by choosing larger finite fields one can alleviate the tradeoff between the length of the linear equivalent and the order of correlation immunity. This is useful for the design of various cryptosystems