The generator matrix

 1  1
 0 X^2+X

generates a code of length 2 over Z2[X]/(X^3) who´s minimum homogenous weight is 1.

Homogenous weight enumerator: w(x)=1x^0+4x^1+22x^2+4x^3+1x^4

The gray image is a linear code over GF(2) with n=8, k=5 and d=2.
As d=2 is an upper bound for linear (8,5,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 5.
This code was found by Heurico 1.16 in 0.456 seconds.