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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

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A linear construction for certain Kerdock and Preparata codes
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by A. R. Calderbank, A. R. Hammons, P. Vijay Kumar, N. J. A. Sloane and Patrick Solé PDF
Bull. Amer. Math. Soc. 29 (1993), 218-222 Request permission

Abstract:

The Nordstrom-Robinson, Kerdock, and (slightly modified) Preparata codes are shown to be linear over ${\mathbb {Z}_4}$, the integers ${\bmod \;4}$. The Kerdock and Preparata codes are duals over ${\mathbb {Z}_4}$, and the Nordstrom-Robinson code is self-dual. All these codes are just extended cyclic codes over ${\mathbb {Z}_4}$. This provides a simple definition for these codes and explains why their Hamming weight distributions are dual to each other. First- and second-order Reed-Muller codes are also linear codes over ${\mathbb {Z}_4}$, but Hamming codes in general are not, nor is the Golay code.
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Additional Information
  • © Copyright 1993 American Mathematical Society
  • Journal: Bull. Amer. Math. Soc. 29 (1993), 218-222
  • MSC (2000): Primary 94B05; Secondary 94B15
  • DOI: https://doi.org/10.1090/S0273-0979-1993-00426-9
  • MathSciNet review: 1215307