The generator matrix

 1  1  1  1
 0  X  2 X+2

generates a code of length 4 over Z4[X]/(X^2+2,2X) who´s minimum homogenous weight is 4.

Homogenous weight enumerator: w(x)=1x^0+30x^4+1x^8

The gray image is a code over GF(2) with n=16, k=5 and d=8.
This code was found by Heurico 1.16 in 5.7e-005 seconds.