A 2-adic approach to the analysis of cyclic codes

Journal Article (Journal Article)

This paper describes how 2-adic numbers can be used to analyze the structure of binary cyclic codes and of cyclic codes defined over ℤ , a ≥ 2, the ring of integers modulo 2 . It provides a 2-adic proof of a theorem of McEliece that characterizes the possible Hamming weights that can appear in a binary cyclic code. A generalization of this theorem is derived that applies to cyclic codes over ℤ that are obtained from binary cyclic codes by a sequence of Hensel lifts. This generalization characterizes the number of times a residue modulo 2 appears as a component of an arbitrary codeword in the cyclic code. The limit of the sequence of Hensel lifts is a universal code defined over the 2-adic integers. This code was first introduced by Calderbank and Sloane (1995), and is the main subject of this paper. Binary cyclic codes and cyclic codes over ℤ are obtained from these universal codes by reduction modulo some power of 2. A special case of particular interest is cyclic codes over ℤ that are obtained from binary cyclic codes by means of a single Hensel lift. The binary images of such codes under the Gray isometry include the Kerdock, Preparata, and Delsarte-Goethals codes. These are nonlinear binary codes that contain more codewords than any linear code presently known. Fundamental understanding of the composition of codewords in cyclic codes over ℤ is central to the search for more families of optimal codes. This paper also constructs even unimodular lattices from the Hensel lift of extended binary cyclic codes that are self-dual with all Hamming weights divisible by 4. The Leech lattice arises in this way as do extremal lattices in dimensions 32 through 48. © 1997 IEEE. 2 2 2 4 4 a a a a a

Full Text

Duke Authors

Cited Authors

  • Calderbank, AR; Li, WCW; Poonen, B

Published Date

  • December 1, 1997

Published In

Volume / Issue

  • 43 / 3

Start / End Page

  • 977 - 986

International Standard Serial Number (ISSN)

  • 0018-9448

Digital Object Identifier (DOI)

  • 10.1109/18.568706

Citation Source

  • Scopus