The generator matrix

 1  0  0  1  1  1  1  1
 0  1  0  1  0  2  2  0
 0  0  1  1  1  1  3  0
 0  0  0  2  0  2  0  0
 0  0  0  0  2  2  2  0

generates a code of length 8 over Z4 who´s minimum homogenous weight is 4.

Homogenous weight enumerator: w(x)=1x^0+10x^4+28x^5+32x^6+36x^7+42x^8+36x^9+32x^10+28x^11+10x^12+1x^16

The gray image is a code over GF(2) with n=16, k=8 and d=4.
This code was found by Heurico 1.16 in 0.00039 seconds.