The generator matrix

 1  0  0  0  0  1  1  1  0  1  1
 0  1  0  0  0  0  1  1  1  0  0
 0  0  1  0  0  1  1  0  1  0  0
 0  0  0  1  0  1  0  1  1  0  0
 0  0  0  0  1  1  0  0  0  2  0
 0  0  0  0  0  2  0  0  2  2  0
 0  0  0  0  0  0  2  0  2  0  0
 0  0  0  0  0  0  0  2  2  0  0

generates a code of length 11 over Z4 who´s minimum homogenous weight is 4.

Homogenous weight enumerator: w(x)=1x^0+70x^4+370x^6+1276x^8+2354x^10+2454x^12+1190x^14+419x^16+54x^18+4x^20

The gray image is a code over GF(2) with n=22, k=13 and d=4.
This code was found by Heurico 1.16 in 0.131 seconds.