The generator matrix

 1  0  0  0  1  1  1  1  1  0  X  X  1
 0  1  0  0  1  0  X  1  1  X  1  1  0
 0  0  1  0  1  0 X+1  X X+1  1  0 X+1  0
 0  0  0  1  1  1  X X+1  0  0  1  1  0
 0  0  0  0  X  0  X  0  X  0  X  0  0
 0  0  0  0  0  X  0  0  X  X  0  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+40x^8+60x^9+80x^10+116x^11+120x^12+152x^13+154x^14+136x^15+91x^16+44x^17+20x^18+4x^19+4x^20+2x^22

The gray image is a linear code over GF(2) with n=26, k=10 and d=8.
As d=8 is an upper bound for linear (26,10,2)-codes, this code is optimal over Z2[X]/(X^2) for dimension 10.
This code was found by Heurico 1.16 in 0.00829 seconds.