A linear construction for certain Kerdock and Preparata codes

The Nordstrom-Robinson, Kerdock, and (slightly modified) Preparata codes are shown to be linear over Z 4 {\mathbb {Z}_4} , the integers mod 4 {\bmod \;4} . The Kerdock and Preparata codes are duals over Z 4 {\mathbb {Z}_4} , and the Nordstrom-Robinson code is self-dual. All these codes are just extended cyclic codes over Z 4 {\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 Z 4 {\mathbb {Z}_4} , but Hamming codes in general are not, nor is the Golay code.