The generator matrix
1 1 X 1 1
0 X X 0 0
0 0 X^2 0 0
0 0 0 X^2 0
0 0 0 0 X^2
generates a code of length 5 over Z2[X]/(X^3) who´s minimum homogenous weight is 2.
Homogenous weight enumerator: w(x)=1x^0+9x^2+86x^4+64x^5+86x^6+9x^8+1x^10
The gray image is a linear code over GF(2) with n=20, k=8 and d=4.
This code was found by Heurico 1.16 in 9.39e-005 seconds.