The generator matrix

1 0 0 0 0 0 1 1 1 1
0 1 0 0 0 0 1 0 0 X
0 0 1 0 0 0 1 0 X 0
0 0 0 1 0 0 1 1 1 X+1
0 0 0 0 1 0 1 X+1 1 0
0 0 0 0 0 1 1 X+1 1 X
0 0 0 0 0 0 X X 0 0

generates a code of length 10 over Z2[X]/(X^2) who´s minimum homogenous weight is 4.

Homogenous weight enumerator: w(x)=1x^0+68x^4+104x^5+294x^6+584x^7+1002x^8+1360x^9+1370x^10+1360x^11+988x^12+584x^13+314x^14+104x^15+53x^16+6x^18

The gray image is a linear code over GF(2) with n=20, k=13 and d=4.
As d=4 is an upper bound for linear (20,13,2)-codes, this code is optimal over Z2[X]/(X^2) for dimension 13.
This code was found by Heurico 1.10 in 0.015 seconds.