On rank and kernel of some mixed perfect codes

Mixed perfect 1-error correcting codes and the associated dual codes over the group Z(n,l)Z(n,l), Z(n,l)=Z2×Z2×⋯×Z2︸n×Z2l,n≥1 and l≥2, are investigated.A lower and an upper bound for the rank kk of the kernel of mixed perfect 1-error correcting codes in Z(n,l)Z(n,l), depending on the rank rr of the mixed perfect code and the structure of the corresponding dual code, are given.Due to a general construction of mixed perfect 1-error correcting group codes in Z(n,l)Z(n,l), we show that the upper bound is tight for some nn, ll and rr.