The generator matrix

 1  1  1  1  1
 0  X  0  X  0
 0  0  X  X  0
 0  0  0 X^2  0
 0  0  0  0 X^2

generates a code of length 5 over Z2[X]/(X^3) who´s minimum homogenous weight is 2.

Homogenous weight enumerator: w(x)=1x^0+17x^2+24x^3+46x^4+336x^5+46x^6+24x^7+17x^8+1x^10

The gray image is a linear code over GF(2) with n=20, k=9 and d=4.
This code was found by Heurico 1.16 in 0.000178 seconds.