Abstract :
We suggest the following construction of codes. In the Hamming code Hn, we first choose a collection of~m disjoint iq-components, q=1,…,m. Then, for each q, we inverse the iqth coordinate in all vectors of the iq-component of this collection. The family of perfect codes obtained in such a way contains Vasilʹevʹs codes and also a number of other codes, possessing various interesting properties: the nonsystematic codes, the codes of full rank, and the codes with trivial automorphism group. In the present article we enumerate all perfect codes of length 15 obtained by means of such a construction from the Hamming code H15; their number is shown to be equal to 131224492.
