On optimal binary one-error-correcting codes of lengths $2^{m}-4$ and $2^{m}-3$

On optimal binary one-error-correcting codes of lengths $2^{m}-4$ and $2^{m}-3$ Full article

Journal

IEEE Transactions on Information Theory
ISSN: 0018-9448 , E-ISSN: 1557-9654

Output data

Year: 2011, Volume: 57, Number: 10, Pages: 6771-6779 Pages count : 9 DOI: 10.1109/tit.2011.2147758

Tags

Automorphism group, classification, clique, error-correcting code, MacWilliams transform

Authors

Krotov D.S. ^1,2 , Östergård P.R.J. ^3,4 , Pottonen O. ^3,5

Affiliations

1	Sobolev Institute of Mathematics
2	Mechanics and Mathematics Department, Novosibirsk State University
3	Department of Communications and Networking, Aalto University School of Electrical Engineering, 00076 Aalto, Finland
4	Lehrstuhl für Mathematik II, Universität Bayreuth, 95440 Bayreuth, Germany
5	Departament de Llenguatges i Sistemes Informàtics, Universitat Politècnica de Catalunya, 08034 Barcelona, Spain

Best and Brouwer proved that triply-shortened and doubly-shortened binary Hamming codes (which have length 2^m-4 and 2^m-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 237610 and 117823 such codes, respectively (with 27375 and 17513 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 2^m-3 that cannot be lengthened to perfect codes of length 2^m-1.

Krotov D.S. , Östergård P.R.J. , Pottonen O.
On optimal binary one-error-correcting codes of lengths $2^{m}-4$ and $2^{m}-3$
IEEE Transactions on Information Theory. 2011. V.57. N10. P.6771-6779. DOI: 10.1109/tit.2011.2147758 WOS Scopus РИНЦ OpenAlex

Submitted:	Feb 18, 2011
Accepted:	Apr 7, 2011
Published online:	May 19, 2011

Web of science:	WOS:000295739000031
Scopus:	2-s2.0-80053959954
Elibrary:	18009028
OpenAlex:	W3123893438

DB	Citing
Web of science	8
Scopus	11
OpenAlex	10