Abstract:
Certain nonlinear binary codes contain more code-words than any comparable linear code presently known. These include the Kerdock and Preparata codes that can be very simply constructed as binary images, under the Gray map, of linear codes over ℤ4 that are defined by means of parity checks involving Galois rings. This paper describes how Fourier transforms on Galois rings and elementary symmetric functions can be used to derive lower bounds on the minimum distance of such codes. These methods and techniques from algebraic geometry are applied to find the exact minimum distance of a family of ℤ4-linear codes with length 2m (m, odd) and size 22m+1-5m-2. The Gray image of the code of length 32 is the best (64, 237) code that is presently known. This paper also determines the exact minimum Lee distance of the linear codes over ℤ4 that are obtained from the extended binary two- and three-error-correcting BCH codes by Hensel lifting. The Gray image of the Hensel lift of the three-error-correcting BCH code of length 32 is the best (64, 232) code that is presently known. This code also determines an extremal 32-dimensional even unimodular lattice. © 1996 IEEE.
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