The Perfect Binary One-Error-Correcting Codes of Length 15: Part II--Properties

TL;DR

This paper analyzes classified perfect binary one-error-correcting codes of length 15, detailing their properties, relationships with Steiner systems, and transformations from the Hamming code, enhancing understanding of their structure and classifications.

Contribution

It provides a detailed study of classified codes, including properties, relationships with Steiner systems, and methods to generate codes via switching from the Hamming code.

Findings

33 of 80 Steiner systems occur in these codes

All but two full-rank codes derive from the Hamming code via switching

Classification of certain mixed perfect codes achieved

Abstract

A complete classification of the perfect binary one-error-correcting codes of length 15 as well as their extensions of length 16 was recently carried out in [P. R. J. \"Osterg{\aa}rd and O. Pottonen, "The perfect binary one-error-correcting codes of length 15: Part I--Classification," IEEE Trans. Inform. Theory vol. 55, pp. 4657--4660, 2009]. In the current accompanying work, the classified codes are studied in great detail, and their main properties are tabulated. The results include the fact that 33 of the 80 Steiner triple systems of order 15 occur in such codes. Further understanding is gained on full-rank codes via switching, as it turns out that all but two full-rank codes can be obtained through a series of such transformations from the Hamming code. Other topics studied include (non)systematic codes, embedded one-error-correcting codes, and defining sets of codes. A classification of certain mixed perfect codes is also obtained.