The generator matrix

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

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+24x^4+80x^5+20x^6+3x^8

The gray image is a linear code over GF(2) with n=40, k=7 and d=16.
As d=18 is an upper bound for linear (40,7,2)-codes, this code is optimal over Z2[X]/(X^4) for dimension 7.
This code was found by Heurico 1.16 in -1.01e-007 seconds.