A linear construction for certain Kerdock and Preparata codes

The Nordstrom-Robinson, Kerdock, and (slightly modified) Preparata codes are shown to be linear over <!-- MATH ${\mathbb{Z}_4}$ --> , the integers <!-- MATH ${\bmod\;4}$ --> . The Kerdock and Preparata codes are duals over <!-- MATH ${\mathbb{Z}_4}$ --> , and the Nordstrom-Robinson code is self-dual. All these codes are just extended cyclic codes over <!-- MATH ${\mathbb{Z}_4}$ --> . This provides a simple definition for these codes and explains why their Hamming weight distributions are dual to each other. First- and second-order Reed-Muller codes are also linear codes over <!-- MATH ${\mathbb{Z}_4}$ --> , but Hamming codes in general are not, nor is the Golay code.