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A linear construction for certain Kerdock and Preparata codes

A. R. Calderbank·A. Roger Hammons Jr.·P. Vijay Kumar·N. J. A. Sloane·Patrick Solé
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Abstract

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.

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A. R. Calderbank, A. Roger Hammons Jr., P. Vijay Kumar, N. J. A. Sloane, Patrick Solé. 1993-10-01. A linear construction for certain Kerdock and Preparata codes. https://arxiv.org/abs/math/9310227

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