The Perfect Binary One-Error-Correcting Codes of Length 15: Part I--Classification
Patric R. J. \"Osterg{\aa}rd, Olli Pottonen
TL;DR
This paper provides a complete classification of perfect binary one-error-correcting codes of length 15 and their extensions, revealing thousands of inequivalent codes and leveraging Steiner quadruple systems for efficient generation.
Contribution
It offers the first comprehensive classification of perfect binary one-error-correcting codes of length 15 and their extensions, including related codes of length 14.
Findings
5983 inequivalent perfect codes of length 15
2165 extended perfect codes of length 16
Classified 38408 codes of length 14
Abstract
A complete classification of the perfect binary one-error-correcting codes of length 15 as well as their extensions of length 16 is presented. There are 5983 such inequivalent perfect codes and 2165 extended perfect codes. Efficient generation of these codes relies on the recent classification of Steiner quadruple systems of order 16. Utilizing a result of Blackmore, the optimal binary one-error-correcting codes of length 14 and the (15, 1024, 4) codes are also classified; there are 38408 and 5983 such codes, respectively.
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