The generator matrix

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

generates a code of length 13 over Z2[X]/(X^2) who´s minimum homogenous weight is 8.

Homogenous weight enumerator: w(x)=1x^0+119x^8+336x^10+544x^12+608x^14+343x^16+80x^18+16x^20+1x^24

The gray image is a linear code over GF(2) with n=26, k=11 and d=8.
As d=8 is an upper bound for linear (26,11,2)-codes, this code is optimal over Z2[X]/(X^2) for dimension 11.
This code was found by Heurico 1.10 in 0.016 seconds.