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) = under(under(Z₂ × Z₂ × ⋯ × Z₂, {presentation form for vertical right curly bracket}), n) × underover(Z, 2, l), n ≥ 1 and l ≥ 2, are investigated. A lower 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. Due to 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.