The Z4-linearity of Kerdock, Preparata, Goethals, and related codes

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 Z4, the integers mod 4. The construction implies that all these binary codes are distance invariant. Duality in the Z4 domain implies that the binary images have dual weight distributions. The Kerdock and Preparata codes are duals over Z4 - 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 Z4-analogs of first-order Reed-Muller and extended Hamming codes, respectively. All these codes are extended cyclic codes over Z4, 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 Z4, but extended Hamming codes of length n greater than 32 and the Golay code are not. Using Z4-linearity, a new family of distance regular graphs are constructed on the cosets of the Preparata code. (Author)