The generator matrix

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

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+130x^8+160x^10+160x^12+320x^14+181x^16+32x^18+40x^20

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 0.00964 seconds.