The generator matrix

 1  1  X  1  1
 0 X^3+X^2 X^3+X^2 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+11x^4+40x^5+10x^6+2x^8

The gray image is a linear code over GF(2) with n=40, k=6 and d=16.
As d=18 is an upper bound for linear (40,6,2)-codes, this code is optimal over Z2[X]/(X^4) for dimension 6.
This code was found by Heurico 1.16 in -3.24e-008 seconds.