The Z_4-Linearity of Kerdock, Preparata, Goethals and Related Codes(*) A. Roger Hammons, Jr.(**) Hughes Aircraft Company Network Systems Division, Germantown, MD 20876 U.S.A. P. Vijay Kumar(**) Communication Science Institute, EE-Systems University of Southern California, Los Angeles, CA 90089~U.S.A. A. R. Calderbank(***) and N. J. A. Sloane(***) Mathematical Sciences Research Center AT&T Bell Laboratories, Murray Hill, NJ 07974 U.S.A. Patrick Sol'{e}(****) CNRS -- I3S, 250 rue A. Einstein, b^{a}timent 4 Sophia -- Antipolis, 06560 Valbonne, France 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. For the full version see http://www.research.att.com/~njas/doc/linear.pdf (pdf) or http://www.research.att.com/~njas/doc/linear.ps (ps) (*) A different version of this paper appeared in IEEE Trans. Inform. Theory, 40 (1994), 301-319. (**) The work of A.~R. Hammons, Jr. and P.~V. Kumar was supported in part by the National Science Foundation under Grant NCR-9016077 and by Hughes Aircraft Company under its Ph.D. fellowship program.~~} (***) Present address: Information Sciences Research AT&T Shannon Lab Florham Park, NJ 07932-0971 USA Email: njas@research.att.com (****) P. Sol'{e} thanks the DIMACS Center and the IEEE for travel support.