CYCLIC CODES OVER Z(4), LOCATOR POLYNOMIALS, AND NEWTONS IDENTITIES

Certain nonlinear binary codes contain more codewords than any compara ble linear code presently known. These include the Kerdock and Prepara ta codes that can be very simply constructed as binary images, under t he Gray map, of linear codes over Z(4) that are defined by means of pa rity checks involving Galois rings. This paper describes how Fourier t ransforms on Galois rings and elementary symmetric functions can be us ed to derive lower bounds on the minimum distance of such codes. These methods and techniques from algebraic geometry are applied to find th e exact minimum distance of a family of Z(4)-linear codes with length 2(m) (m, odd) and size 2(2m+1-5m-2). The Gray image of the code of len gth 32 is the best (64, 2(37)) code that is presently known. This pape r also determines the exact minimum Lee distance of the linear codes o ver Z(4) that are obtained from the extended binary two- and three-err or-correcting BCH codes by Hensel lifting. The Gray image of the Hense l lift of the three-error-correcting BCH code of length 32 is the best (64, 2(32)) code that is presently known. This code also determines a n extremal 32-dimensional even unimodular lattice.