On optimal binary one-error-correcting codes of lengths 2m-4 and 2m-3

Krotov, Denis S., Ostergard, Patric R. J. and Pottonen, Olli (2011) On optimal binary one-error-correcting codes of lengths 2m-4 and 2m-3. IEEE Transactions on Information Theory, 57 10: 6771-6779. doi:10.1109/TIT.2011.2147758

Author Krotov, Denis S.
Ostergard, Patric R. J.
Pottonen, Olli
Title On optimal binary one-error-correcting codes of lengths 2m-4 and 2m-3
Journal name IEEE Transactions on Information Theory   Check publisher's open access policy
ISSN 0018-9448
Publication date 2011-01-01
Year available 2011
Sub-type Article (original research)
DOI 10.1109/TIT.2011.2147758
Open Access Status Not Open Access
Volume 57
Issue 10
Start page 6771
End page 6779
Total pages 9
Place of publication Piscataway, NJ United States
Publisher Institute of Electrical and Electronics Engineers
Language eng
Subject 1710 Information Systems
1706 Computer Science Applications
3309 Library and Information Sciences
Abstract Best and Brouwer proved that triply-shortened and doubly-shortened binary Hamming codes (which have length 2m-4 and 2m-3, respectively) are optimal. Properties of such codes are here studied, determining among other things parameters of certain subcodes. A utilization of these properties makes a computer-aided classification of the optimal binary one-error-correcting codes of lengths 12 and 13 possible; there are 237 610 and 117 823 such codes, respectively (with 27 375 and 17 513 inequivalent extensions). This completes the classification of optimal binary one-error-correcting codes for all lengths up to 15. Some properties of the classified codes are further investigated. Finally, it is proved that for any m ≥ 4, there are optimal binary one-error-correcting codes of length 2 m-4 and 2m-3 that cannot be lengthened to perfect codes of length 2m-1.
Keyword Automorphism group
Error correcting code
MacWilliams transform
Q-Index Code C1
Q-Index Status Provisional Code
Institutional Status Non-UQ

Document type: Journal Article
Sub-type: Article (original research)
Collection: School of Mathematics and Physics
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