The generator matrix

 1  0  0  1  1  1  X  1  1  1  1
 0  1  0  0  1 X^2+X+1  1  X X^2+X X^2  0
 0  0  1 X+1 X^2+X+1  X  1  1 X+1 X+1 X^2+1
 0  0  0 X^2  0 X^2 X^2  0  0 X^2  0
 0  0  0  0 X^2 X^2 X^2 X^2  0 X^2 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+186x^8+192x^9+288x^10+640x^11+424x^12+192x^13+96x^14+29x^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 2.86 seconds.