Three characterizations of non-binary correlation-immune and resilient functions

A functionf(X₁,X₂, ...,X_n) is said to betth-order correlation-immune if the random variableZ=f(X₁,X₂,...,X_n) is independent of every set oft random variables chosen from the independent equiprobable random variablesX₁,X₂,...,X_n. Additionally, if all possible outputs are equally likely, thenf is called at-resilient function. In this paper, we provide three different characterizations oft th-order correlation immune functions and resilient functions where the random variable is overGF (q). The first is in terms of the structure of a certain associated matrix. The second characterization involves Fourier transforms. The third characterization establishes the equivalence of resilient functions and large sets of orthogonal arrays.