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 (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 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-analogues 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 firstand second-order ReedMuller codes are also linear over Z4, but extended Hamming codes of length n ≥ 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. 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. P. Solé thanks the DIMACS Center and the IEEE for travel support. The Z4-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 Solé CNRS – I3S, 250 rue A. Einstein, bâtiment 4 Sophia – Antipolis, 06560 Valbonne, France