  The Nordstrom-Robinson, Kerdock, and (slightly modified) Pre\- parata codes
are shown to be linear over $\ZZ_4$, the integers $\bmod~4$. The Kerdock and
Preparata codes are duals over $\ZZ_4$, and the Nordstrom-Robinson code is
self-dual. All these codes are just extended cyclic codes over $\ZZ_4$. 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 $\ZZ_4$, but Hamming codes in
general are not, nor is the Golay code.
