The paper explores the $ Z_4 $-linearity of several notable nonlinear binary codes, including the Kerdock, Preparata, Goethals, and Nordstrom-Robinson codes. It demonstrates that these codes can be constructed as binary images under the Gray map of linear codes over $ Z_4 $, the integers modulo 4. This construction implies that these codes are distance-invariant and have dual weight distributions. The Kerdock and Preparata codes are shown to be duals over $ Z_4 $, explaining their dual weight distributions. The Kerdock code is a $ Z_4 $-analogue of the first-order Reed-Muller code, while the Preparata code is a $ Z_4 $-analogue of the extended Hamming code. All these codes are extended cyclic codes over $ Z_4 $, which simplifies encoding and decoding. The paper also presents algebraic hard-decision decoding algorithms for the Preparata and Kerdock codes. It shows that binary first- and second-order Reed-Muller codes are linear over $ Z_4 $, but extended Hamming codes of length $ n \geq 32 $ and the Golay code are not. The paper constructs a new family of distance-regular graphs on the cosets of the Preparata code. The Nordstrom-Robinson code is shown to be a $ Z_4 $-version of the octacode, a self-dual code over $ Z_4 $. The paper also discusses the properties of quaternary codes, their weight enumerators, and the relationship between quaternary and binary codes. It provides a detailed analysis of the structure and properties of these codes, including their duals, weight distributions, and decoding algorithms.The paper explores the $ Z_4 $-linearity of several notable nonlinear binary codes, including the Kerdock, Preparata, Goethals, and Nordstrom-Robinson codes. It demonstrates that these codes can be constructed as binary images under the Gray map of linear codes over $ Z_4 $, the integers modulo 4. This construction implies that these codes are distance-invariant and have dual weight distributions. The Kerdock and Preparata codes are shown to be duals over $ Z_4 $, explaining their dual weight distributions. The Kerdock code is a $ Z_4 $-analogue of the first-order Reed-Muller code, while the Preparata code is a $ Z_4 $-analogue of the extended Hamming code. All these codes are extended cyclic codes over $ Z_4 $, which simplifies encoding and decoding. The paper also presents algebraic hard-decision decoding algorithms for the Preparata and Kerdock codes. It shows that binary first- and second-order Reed-Muller codes are linear over $ Z_4 $, but extended Hamming codes of length $ n \geq 32 $ and the Golay code are not. The paper constructs a new family of distance-regular graphs on the cosets of the Preparata code. The Nordstrom-Robinson code is shown to be a $ Z_4 $-version of the octacode, a self-dual code over $ Z_4 $. The paper also discusses the properties of quaternary codes, their weight enumerators, and the relationship between quaternary and binary codes. It provides a detailed analysis of the structure and properties of these codes, including their duals, weight distributions, and decoding algorithms.