The generator matrix

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

generates a code of length 4 over Z4 who´s minimum homogenous weight is 1.

Homogenous weight enumerator: w(x)=1x^0+8x^1+28x^2+56x^3+70x^4+56x^5+28x^6+8x^7+1x^8

The gray image is a code over GF(2) with n=8, k=8 and d=1.
This code was found by Heurico 1.16 in 6.79e-005 seconds.