  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
$2^m-1$.
