Cyclic codes over ℤ4 locator polynomials, and newton's identities

Journal Article (Journal Article)

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 ℤ 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 ℤ -linear codes with length 2 (m, odd) and size 2 . The Gray image of the code of length 32 is the best (64, 2 ) code that is presently known. This paper also determines the exact minimum Lee distance of the linear codes over ℤ 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, 2 ) code that is presently known. This code also determines an extremal 32-dimensional even unimodular lattice. © 1996 IEEE. 4 4 4 m 2m+1-5m-2 37 32

Full Text

Duke Authors

Cited Authors

  • Calderbank, AR; McGuire, G; Kumar, PV; Helleseth, T

Published Date

  • December 1, 1996

Published In

Volume / Issue

  • 42 / 1

Start / End Page

  • 217 - 226

International Standard Serial Number (ISSN)

  • 0018-9448

Digital Object Identifier (DOI)

  • 10.1109/18.481791

Citation Source

  • Scopus