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

TL;DR

This paper studies properties and classifications of optimal binary one-error-correcting codes of lengths $2^m-4$ and $2^m-3$, completing the classification for lengths up to 15 and exploring their extension properties.

Contribution

It provides a detailed analysis of these codes, classifies all optimal codes of lengths 12 and 13, and proves the existence of codes that cannot be extended to perfect codes.

Findings

Classified all optimal codes of lengths 12 and 13.

Determined parameters of subcodes within these codes.

Proved some codes cannot be extended to perfect codes.

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 $2^m-1$.