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

Author

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

Author_Institution

Dept. of Commun. & Networking, Aalto Univ., Aalto, Finland

Volume

56

Issue

6

fYear

2010

fDate

6/1/2010 12:00:00 AM

Firstpage

2571

Lastpage

2582

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. OÂ¿stergÃ¥rd and O. Pottonen, Â¿The perfect binary one-error-correcting codes of length 15: Part I-Classification,Â¿ 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.

Keywords

Hamming codes; binary codes; error correction codes; Hamming code; Steiner triple system; binary one-error-correcting code; embedded one-error-correcting code; mixed perfect code; systematic code; Automation; Binary codes; Electronic mail; Galois fields; Kernel; Mathematics; Statistics; Classification; Hamming code; Steiner system; perfect code; switching;

fLanguage

English

Journal_Title

Information Theory, IEEE Transactions on

Publisher

ieee

ISSN

0018-9448

Type

jour

DOI

10.1109/TIT.2010.2046197

Filename

5466538