The generator matrix

 1  1  1  1 X^2
 0  X X^3+X^2 X^2+X X^3+X^2
 0  0 X^3 X^3  0

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

Homogenous weight enumerator: w(x)=1x^0+98x^4+64x^5+88x^6+5x^8

The gray image is a linear code over GF(2) with n=40, k=8 and d=16.
As d=16 is an upper bound for linear (40,8,2)-codes, this code is optimal over Z2[X]/(X^4) for dimension 8.
This code was found by Heurico 1.16 in 1.05e-007 seconds.