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The Z_4-Linearity of Kerdock, Preparata, Goethals and Related Codes

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

Certain notorious nonlinear binary codes contain more codewords than any known linear code. These include the codes constructed by Nordstrom-Robinson, Kerdock, Preparata, Goethals, and Delsarte-Goethals. It is shown here that all these codes can be very simply constructed as binary images under the Gray map of linear codes over Z_4, the integers mod 4 (although this requires a slight modification of the Preparata and Goethals codes). The construction implies that all these binary codes are distance invariant. Duality in the Z_4 domain implies that the binary images have dual weight distributions. The Kerdock and "Preparata" codes are duals over Z_4 -- and the Nordstrom-Robinson code is self-dual -- which explains why their weight distributions are dual to each other. The Kerdock and "Preparata" codes are Z_4-analogues of first-order Reed-Muller and extended Hamming codes, respectively. All these codes are extended cyclic codes over Z_4, which greatly simplifies encoding and decoding. An algebraic hard-decision decoding algorithm is given for the "Preparata" code and a Hadamard-transform soft-decision decoding algorithm for the Kerdock code. Binary first- and second-order Reed-Muller codes are also linear over Z_4, but extended Hamming codes of length n >= 32 and the Golay code are not. Using Z_4-linearity, a new family of distance regular graphs are constructed on the cosets of the "Preparata" code.

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A. Roger Hammons, Jr., P. Vijay Kumar, A. R. Calderbank, N. J. A. Sloane, Patrick Solé. 2002-07-23. The Z_4-Linearity of Kerdock, Preparata, Goethals and Related Codes. https://arxiv.org/abs/math/0207208

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