The generator matrix

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

generates a code of length 5 over Z4 who´s minimum homogenous weight is 2.

Homogenous weight enumerator: w(x)=1x^0+17x^2+24x^3+46x^4+80x^5+46x^6+24x^7+17x^8+1x^10

The gray image is a code over GF(2) with n=10, k=8 and d=2.
This code was found by Heurico 1.16 in 0.000121 seconds.