In analogy to the corresponding measures of pseudorandomness for quaternary sequences introduced by Mauduit and S\'ark\"ozy (for $m$-ary sequences) we introduce the well-distribution measure and correlation measure of order $k$ for sequences over $\mathbb{F}_4$. Using any fixed bijection from $\mathbb{F}_4$ to the set of complex fourth roots of unity, we analyze the relation of these pseudorandomness measures for sequences over $\mathbb{F}_4$ and for the corresponding quaternary sequences. More precisely, we show that they differ only by a multiplicative constant (depending only on $k$). We also apply the results for deriving new quaternary pseudorandom sequences from pseudorandom sequences over $\mathbb{F}_4$ and vice versa.

This is joint work with Arne Winterhof.