The nonlinearity of the-Z2-Kerdock code Km+1, where m is an odd integer ≥3, is not at all obvious. By regarding Km+1 as the binary image of the-Z4-Kerdock code K (m) under the Gray map[1], a simple proof can be achieved (cf. Theorems 8.7 and 8.9 of [2]). Similarly, the formal duality of the-Z2-Kerdock code Km+1 and the-Z2-Preparata code Pm+1 is even more not at all obvious. A simple proof is given in the present note, by regarding Km+1 as the binary image of K (m) and using the duality of K (m) and the-Z4-Preparata code P (m) established in [1].