The generator matrix

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

generates a code of length 11 over Z2[X]/(X^2) who´s minimum homogenous weight is 8.

Homogenous weight enumerator: w(x)=1x^0+34x^8+24x^10+48x^12+8x^14+13x^16

The gray image is a linear code over GF(2) with n=22, k=7 and d=8.
As d=8 is an upper bound for linear (22,7,2)-codes, this code is optimal over Z2[X]/(X^2) for dimension 7.
This code was found by Heurico 1.16 in 0.000637 seconds.