The generator matrix

 1  0  0  1  1  1  1  0  1  X  X
 0  1  0  1  0  1  X  1  X  1  1
 0  0  1  1  1  X  X X+1 X^2  0  0
 0  0  0  X X^2  0 X^2+X X^2+X X^2 X^2  X

generates a code of length 11 over Z2[X]/(X^3) who´s minimum homogenous weight is 8.

Homogenous weight enumerator: w(x)=1x^0+180x^8+120x^9+578x^10+280x^11+612x^12+104x^13+156x^14+8x^15+7x^16+2x^18

The gray image is a linear code over GF(2) with n=44, k=11 and d=16.
As d=17 is an upper bound for linear (44,11,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 11.
This code was found by Heurico 1.11 in -3.24e-008 seconds.