On Optimal Binary One-Error-Correcting Codes of Lengths $2^{m}-4$ and $2^{m}-3$

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 237 $\thinspace$610 and 117 $\thinspace$823 such codes, respectively (with 27$\thinspace$ 375 and 17$\thinspace$ 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 $m \geq 4$, 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 $2^{m}-1$.