The generator matrix

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

generates a code of length 14 over Z4 who´s minimum homogenous weight is 4.

Homogenous weight enumerator: w(x)=1x^0+63x^4+56x^6+509x^8+288x^10+1987x^12+2384x^14+1987x^16+288x^18+509x^20+56x^22+63x^24+1x^28

The gray image is a code over GF(2) with n=28, k=13 and d=4.
This code was found by Heurico 1.16 in 0.144 seconds.