The generator matrix
1 1 1 1 1
0 X 0 0 X^2
0 0 X 0 0
0 0 0 X 0
0 0 0 0 X
generates a code of length 5 over Z2[X]/(X^3) who´s minimum homogenous weight is 1.
Homogenous weight enumerator: w(x)=1x^0+10x^1+45x^2+120x^3+210x^4+1276x^5+210x^6+120x^7+45x^8+10x^9+1x^10
The gray image is a linear code over GF(2) with n=20, k=11 and d=2.
This code was found by Heurico 1.16 in 0.000986 seconds.