The generator matrix
1 1 X X X
0 X^3 0 X^3 0
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+8x^4+48x^5+4x^6+3x^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 -1.01e-007 seconds.