The paper discusses the $\mathbb{Z}_4$-linearity of several nonlinear binary codes, including the Nordstrom-Robinson, Kerdock, Preparata, Goethals, and Delsarte-Goethals codes. It shows that these codes can be constructed as binary images of linear codes over $\mathbb{Z}_4$ under the Gray map. The construction implies that these binary codes are distance-invariant, and their dual weight distributions are related through the MacWilliams transform. The Kerdock and Preparata codes are shown to be duals over $\mathbb{Z}_4$, with the Kerdock code being a $\mathbb{Z}_4$-analogue of first-order Reed-Muller codes and the Preparata code being a $\mathbb{Z}_4$-analogue of extended Hamming codes. The paper also provides algebraic decoding algorithms for the Preparata and Kerdock codes and discusses the automorphism groups and distance-regular graphs associated with these codes.The paper discusses the $\mathbb{Z}_4$-linearity of several nonlinear binary codes, including the Nordstrom-Robinson, Kerdock, Preparata, Goethals, and Delsarte-Goethals codes. It shows that these codes can be constructed as binary images of linear codes over $\mathbb{Z}_4$ under the Gray map. The construction implies that these binary codes are distance-invariant, and their dual weight distributions are related through the MacWilliams transform. The Kerdock and Preparata codes are shown to be duals over $\mathbb{Z}_4$, with the Kerdock code being a $\mathbb{Z}_4$-analogue of first-order Reed-Muller codes and the Preparata code being a $\mathbb{Z}_4$-analogue of extended Hamming codes. The paper also provides algebraic decoding algorithms for the Preparata and Kerdock codes and discusses the automorphism groups and distance-regular graphs associated with these codes.