The generator matrix

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

generates a code of length 7 over Z4 who´s minimum homogenous weight is 4.

Homogenous weight enumerator: w(x)=1x^0+26x^4+32x^5+37x^6+64x^7+37x^8+32x^9+26x^10+1x^14

The gray image is a code over GF(2) with n=14, k=8 and d=4.
This code was found by Heurico 1.16 in 0.000361 seconds.