DIMACS TR: 93-52

The Z sub 4-Linearity of Kerdock, Preparata, Goethals and Related Codes

Authors: A.Roger Hammons, Jr., P.Vijay Kumar, A. R. Calderbank, N. J. A. Sloane, and Patrick Sole


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 sub 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 sub 4 domain implies that the binary images have dual weight distributions. The Kerdock and `Preparata' codes are duals over Z sub 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 sub 4-analogues of first-order Reed-Muller and extended Hamming codes, respectively. All these codes are extended cyclic codes over Z sub 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 sub 4, but extended Hamming codes of length n >= 32 and the Golay code are not. Using Z sub 4-linearity, a new family of distance regular graphs are constructed on the cosets of the `Preparata' code.

Paper available at: ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/1993/93-52.ps
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