The generator matrix

 1  1  1  1  1  1  1  X  X  X  0  0  0
 0  X  0  X  0  0  X  0  X  X  X  X  0
 0  0  X  X  0  X  X  X  X  0  0  X  X

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

Homogenous weight enumerator: w(x)=1x^0+8x^13+4x^14+3x^16

The gray image is a linear code over GF(2) with n=26, k=4 and d=13.
As d=13 is an upper bound for linear (26,4,2)-codes, this code is optimal over Z2[X]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in 0.000451 seconds.