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A 2-adic approach to the analysis of cyclic codes

Publication ,  Journal Article
Calderbank, AR; Li, WCW; Poonen, B
Published in: IEEE Transactions on Information Theory
December 1, 1997

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 2a, a ≥ 2, the ring of integers modulo 2 a. 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 2a that are obtained from binary cyclic codes by a sequence of Hensel lifts. This generalization characterizes the number of times a residue modulo 2 a 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 2aare obtained from these universal codes by reduction modulo some power of 2. A special case of particular interest is cyclic codes over ℤ 4 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 4 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.

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Published In

IEEE Transactions on Information Theory

DOI

ISSN

0018-9448

Publication Date

December 1, 1997

Volume

43

Issue

3

Start / End Page

977 / 986

Related Subject Headings

  • Networking & Telecommunications
  • 1005 Communications Technologies
  • 0906 Electrical and Electronic Engineering
  • 0801 Artificial Intelligence and Image Processing
 

Citation

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Calderbank, A. R., Li, W. C. W., & Poonen, B. (1997). A 2-adic approach to the analysis of cyclic codes. IEEE Transactions on Information Theory, 43(3), 977–986. https://doi.org/10.1109/18.568706
Calderbank, A. R., W. C. W. Li, and B. Poonen. “A 2-adic approach to the analysis of cyclic codes.” IEEE Transactions on Information Theory 43, no. 3 (December 1, 1997): 977–86. https://doi.org/10.1109/18.568706.
Calderbank AR, Li WCW, Poonen B. A 2-adic approach to the analysis of cyclic codes. IEEE Transactions on Information Theory. 1997 Dec 1;43(3):977–86.
Calderbank, A. R., et al. “A 2-adic approach to the analysis of cyclic codes.” IEEE Transactions on Information Theory, vol. 43, no. 3, Dec. 1997, pp. 977–86. Scopus, doi:10.1109/18.568706.
Calderbank AR, Li WCW, Poonen B. A 2-adic approach to the analysis of cyclic codes. IEEE Transactions on Information Theory. 1997 Dec 1;43(3):977–986.

Published In

IEEE Transactions on Information Theory

DOI

ISSN

0018-9448

Publication Date

December 1, 1997

Volume

43

Issue

3

Start / End Page

977 / 986

Related Subject Headings

  • Networking & Telecommunications
  • 1005 Communications Technologies
  • 0906 Electrical and Electronic Engineering
  • 0801 Artificial Intelligence and Image Processing