The generator matrix

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

generates a code of length 13 over Z2[X]/(X^2) who´s minimum homogenous weight is 10.

Homogenous weight enumerator: w(x)=1x^0+63x^10+68x^12+67x^14+43x^16+13x^18+1x^22

The gray image is a linear code over GF(2) with n=26, k=8 and d=10.
As d=10 is an upper bound for linear (26,8,2)-codes, this code is optimal over Z2[X]/(X^2) for dimension 8.
This code was found by Heurico 1.16 in 0.00356 seconds.