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 Za, 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

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... ce 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 Z* -analogues of first-order Reed-Muller and extended Hamming codes, respectively. All these codes are extended cyclic codes over Z4, which greatly simplies encoding and decoding. An algebraic hard-decision decoding algorithm is given for the "Preparata" code and a Hadamard-transform softdecision decoding algorithm for the Kerdock code. Binary firstand second-order Reed-Muller codes are also linear over &, but extended Hamming codes of length n >_ 32 and the Golay code are not. Using Z.$-linearity, a new family of distance regular graphs are constructed on the cosets of the "Preparata" code.