The generator matrix

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

generates a code of length 11 over Z2[X]/(X^3) who´s minimum homogenous weight is 10.

Homogenous weight enumerator: w(x)=1x^0+14x^10+32x^11+15x^12+1x^14+1x^18

The gray image is a linear code over GF(2) with n=44, k=6 and d=20.
As d=21 is an upper bound for linear (44,6,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 6.
This code was found by Heurico 1.16 in 0.000219 seconds.