Computer Science – Information Theory
Scientific paper
2011-04-20
IEEE Trans. Inf. Theory 57(10) 2011, 6771-6779
Computer Science
Information Theory
Accepted for publication in IEEE Transactions on Information Theory. Data available at http://www.iki.fi/opottone/codes
Scientific paper
10.1109/TIT.2011.2147758
Best and Brouwer [Discrete Math. 17 (1977), 235-245] 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 $m \geq 4$, 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 Denis S.
Östergård Patric R. J.
Pottonen Olli
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