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    On Optimal Binary One-Error-Correcting Codes of Lengths 2m−42^m-4 and 2m−32^m-3

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    Best and Brouwer [Discrete Math. 17 (1977), 235-245] proved that triply-shortened and doubly-shortened binary Hamming codes (which have length 2m−42^m-4 and 2m−32^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≥4m \geq 4, there are optimal binary one-error-correcting codes of length 2m−42^m-4 and 2m−32^m-3 that cannot be lengthened to perfect codes of length 2m−12^m-1.Comment: Accepted for publication in IEEE Transactions on Information Theory. Data available at http://www.iki.fi/opottone/code
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