The generator matrix

 1  1  X  1  1
 0  X  X  0  0
 0  0 X^2  0  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+9x^2+86x^4+64x^5+86x^6+9x^8+1x^10

The gray image is a linear code over GF(2) with n=20, k=8 and d=4.
This code was found by Heurico 1.16 in 9.39e-005 seconds.