The generator matrix

 1  1  1  1
 0 12  0  8
 0  0 12  0
 0  0  0  4

generates a code of length 4 over Z16 who´s minimum homogenous weight is 1.

Homogenous weight enumerator: w(x)=1x^0+8x^1+28x^2+56x^3+838x^4+56x^5+28x^6+8x^7+1x^8

The gray image is a code over GF(2) with n=32, k=10 and d=4.
This code was found by Heurico 1.16 in 0.000168 seconds.