The generator matrix

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

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+362x^8+480x^10+1096x^12+32x^14+77x^16

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 1.16 in 0.281 seconds.