The generator matrix
1 1 1 1 X
0 X 0 X^2+X X^2
0 0 X^2 X^2 X^2
generates a code of length 5 over Z2[X]/(X^3) who´s minimum homogenous weight is 4.
Homogenous weight enumerator: w(x)=1x^0+18x^4+32x^5+8x^6+5x^8
The gray image is a linear code over GF(2) with n=20, k=6 and d=8.
As d=8 is an upper bound for linear (20,6,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 6.
This code was found by Heurico 1.16 in 0.000116 seconds.