The generator matrix

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

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+92x^4+560x^6+2022x^8+2848x^10+2012x^12+560x^14+97x^16

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 1.16 in 0.133 seconds.