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 2 × Z 2 × ⋯ × Z 2 ︸ n × Z 2 l , n ≥ 1  and  l ≥ 2 , are investigated. r and an upper bound for the rank k of the kernel of mixed perfect 1-error correcting codes in Z ( n , l ) , depending on the rank r of the mixed perfect code and the structure of the corresponding dual code, are given. a general construction of mixed perfect 1-error correcting group codes in Z ( n , l ) , we show that the upper bound is tight for some n , l and r .