The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+8x^11+5x^12+2x^14

The gray image is a linear code over GF(2) with n=44, k=4 and d=22.
As d=23 is an upper bound for linear (44,4,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 4.
This code was found by Heurico 1.16 in 0.00029 seconds.