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