The generator matrix

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

generates a code of length 10 over Z2[X]/(X^3) who´s minimum homogenous weight is 10.

Homogenous weight enumerator: w(x)=1x^0+28x^10+2x^12+1x^16

The gray image is a linear code over GF(2) with n=40, k=5 and d=20.
As d=20 is an upper bound for linear (40,5,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 5.
This code was found by Heurico 1.16 in 0.000245 seconds.