The Z_4-Linearity of Kerdock, Preparata, Goethals and Related Codes

TL;DR

This paper reveals that several nonlinear binary codes, including Kerdock and Preparata, are actually images of linear codes over Z_4, providing new insights into their structure, duality, and decoding methods.

Contribution

It demonstrates that these nonlinear codes are Z_4-linear, simplifies their encoding/decoding, and introduces new algebraic and graph-theoretic constructions.

Findings

Codes are Z_4-analogues of Reed-Muller and Hamming codes.

Provides algebraic decoding algorithms for Preparata and Kerdock codes.

Constructs new distance regular graphs from Preparata code cosets.

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.