The perfect binary one-error-correcting codes of length 15: part II-properties

  • Authors:

  • Patric R. J. Östergård;Olli Pottonen;Kevin T. Phelps

  • Affiliations:

  • Department of Communications and Networking, Aalto University, Aalto, Finland;Finnish Defence Forces Technical Research Centre, Riihimäki, Finland and Department of Communications and Networking, Aalto University, Aalto, Finland;Department of Mathematics and Statistics, Auburn University, Auburn, AL

  • Venue:
  • IEEE Transactions on Information Theory
  • Year:
  • 2010

Quantified Score

Hi-index 754.84

Visualization

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. Östergård and O. Pottonen, "The perfect binary one-error-correcting codes of length 15: Part I--Classification," IEEE Trans. Inf. 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.