The generator matrix

 1  0  0  0  0  1  1  1  1  1
 0  1  0  0  0  0  0  1  3  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

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

Homogenous weight enumerator: w(x)=1x^0+100x^4+528x^6+2062x^8+2848x^10+1972x^12+592x^14+89x^16

The gray image is a code over GF(2) with n=20, k=13 and d=4.
This code was found by Heurico 1.16 in 0.129 seconds.