The generator matrix

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

generates a code of length 5 over Z4 who´s minimum homogenous weight is 1.

Homogenous weight enumerator: w(x)=1x^0+10x^1+45x^2+120x^3+210x^4+252x^5+210x^6+120x^7+45x^8+10x^9+1x^10

The gray image is a code over GF(2) with n=10, k=10 and d=1.
This code was found by Heurico 1.16 in 0.000508 seconds.