The generator matrix

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

generates a code of length 5 over Z2[X]/(X^3) who´s minimum homogenous weight is 4.

Homogenous weight enumerator: w(x)=1x^0+3x^4+24x^5+3x^6+1x^10

The gray image is a linear code over GF(2) with n=20, k=5 and d=8.
As d=9 is an upper bound for linear (20,5,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 5.
This code was found by Heurico 1.16 in 6.2e-005 seconds.