The generator matrix

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

generates a code of length 10 over Z4 who´s minimum homogenous weight is 4.

Homogenous weight enumerator: w(x)=1x^0+189x^4+1120x^6+4034x^8+5696x^10+4034x^12+1120x^14+189x^16+1x^20

The gray image is a code over GF(2) with n=20, k=14 and d=4.
This code was found by Heurico 1.16 in 0.868 seconds.