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

Denis S. Krotov; Patric R. J. Östergård; Olli Pottonen

doi:10.1109/TIT.2011.2147758

paper

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

arXiv:1104.4013 · doi:10.1109/TIT.2011.2147758

Abstract

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 $2^{m} - 1$ .

Accepted for publication in IEEE Transactions on Information Theory. Data available at http://www.iki.fi/opottone/codes

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

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

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