Kernels and p-kernels of pr -ary 1-perfect codes

The rank of a q-ary code C is the dimension of the subspace spanned by C. The kernel of a q-ary code C of length n can be defined as the set of all translations leaving C invariant. Some relations between the rank and the dimension of the kernel of q-ary 1-perfect codes, over script F signq = GF(q) as well as over the prime field script F signp, are established. Q-ary 1-perfect codes of length n=(q m - 1)/(q - 1) with different kernel dimensions using switching constructions are constructed and some upper and lower bounds for the dimension of the kernel, once the rank is given, are established. © 2005 Springer Science+Business Media, Inc.